# Limit theorems for conditional U-statistics analysis on hyperspheres for missing at random data in the presence of measurement error

Salim Bouzebda, Nourelhouda Taachouche

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Salim Bouzebda, Nourelhouda Taachouche. Limit theorems for conditional U-statistics analysis on hyperspheres for missing at random data in the presence of measurement error. Journal of Computational and Applied Mathematics, 2025, 472, pp.116811. 10.1016/j.cam.2025.116811. hal-05089396

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Submitted on 5 Nov 2025

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# Limit theorems for conditional *U*-statistics analysis on hyperspheres for missing at random data in the presence of measurement error

Salim Bouzebda<sup>\*ID</sup>, Nourelhouda Taachouche

Université de Technologie de Compiègne, LMAC, 57 Avenue de Landshut CS 60319, Compiègne, 60203, France

## ARTICLE INFO

MSC:

60F05

60G15

62G08

62G07

62G20

Keywords:

Conditional *U*-statistic

Deconvolution

Errors-in-variables

Nonparametric regression

Manifold-valued data

Lie groups

Measurement errors

Non-euclidean data

Hoeffding decomposition

Density estimation

Missing at random

## ABSTRACT

Missing data and measurement errors are prevalent challenges in modern statistical analyses, mainly when observations lie on complex structures like unit hyperspheres. To address these issues, we introduce a comprehensive framework for conditional *U*-statistics of general order, tailored explicitly for data missing at random and contaminated by measurement errors in such settings. We propose a novel deconvolution method for these conditional *U*-statistics and, for the first time, investigate its convergence rate and asymptotic distribution. Our unified approach establishes general asymptotic properties under broad model conditions, enabling us to derive asymptotic confidence intervals based on the estimator's distribution. To demonstrate the practical significance of our framework, we provide new insights into the Kendall rank correlation coefficient and address discrimination problems.

## 1. Introduction

*U*-statistics, introduced by [1] and building on earlier pioneering work by [2], are a pivotal class of statistical measures in nonparametric statistics. Consider an infinite sequence of random variables {$X_i$}$_{i=1}^\infty$ taking values in a measurable space $(S, \mathcal{S})$. Let $\varphi: S^k \to \mathbb{R}$ be a symmetric and measurable function, known as the kernel of the *U*-statistics, where $k$ is a fixed positive integer representing the order of the statistic. For a sample size $n \ge k$, the *U*-statistic $U_n(\varphi)$ of order $k$ is defined as

$$U_n(\varphi) = \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi(\mathbf{X}_{i_1}, \dots, \mathbf{X}_{i_k}),$$

where $I_n^k$ denotes the set of all $k$-tuples of distinct indices from $\{1, \dots, n\}$:

$$I_n^k = \{(i_1, \dots, i_k) : i_j \in \{1, \dots, n\}, i_j \neq i_\ell \text{ if } j \neq \ell\},$$

and $\mathbb{N}$ represents the set of positive integers. The normalizing factor $\frac{(n-k)!}{n!}$ accounts for the total number of permutations. The statistic $U_n(\varphi)$ serves as the nonparametric uniformly minimum variance unbiased estimator of the parameter $\theta = E[\varphi(\mathbf{X}_1, \dots, \mathbf{X}_k)]$, where

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: salim.bouzebda@utc.fr (S. Bouzebda), nourelhouda.taachouche@outlook.fr (N. Taachouche).</footnote>
the expectation is taken over independent copies of $X_1, \dots, X_k$. It is designed to minimize, with respect to $\alpha$, the sum of squared deviations:

$$ \sum_{1 \le i_1 < \dots < i_k \le n} (\varphi(\mathbf{X}_{i_1}, \dots, \mathbf{X}_{i_k}) - \alpha)^2, $$

as detailed in [3, Theorem 4, Section 1]. [1] initially developed the asymptotic properties of U-statistics, providing the first results on their convergence and limiting distributions. Subsequent research has extensively studied and extended these properties, with significant contributions from [4–7]. Parallel developments for V-statistics, which are closely related to U-statistics but involve averaging over all combinations (including possible repetitions) of the sample, were introduced by [8]. A detailed and rigorous treatment of U-statistics — covering both their fundamental theoretical aspects and a wide range of applications — can be found in the seminal works [3,5,9]. [10] provides a detailed exposition of the underlying theory. U-statistics are relevant in various areas, including testing qualitative features in nonparametric statistics [11,12], cross-validation for density estimation [13], and analyzing limiting distributions of M-estimators [9,14]. In normality testing, [15] introduced tests based on U-processes, while [16] proposed new normality tests utilizing weighted $L_1$-distances and local U-statistics. In machine learning, U-statistics naturally emerge in various applications such as clustering, image recognition, ranking algorithms, and graph-based learning methods. Many empirical risk estimation procedures can be formulated in terms of U-statistics, as shown by [17–19]. U-statistics of infinite order have been used in constructing simultaneous prediction intervals [20]. Further applications include genetic data analysis, where [21] utilized them for model-free clustering and classification, and random compressed sensing explored by [22]. In functional data analysis, [23] studied Kendall’s statistics. [24] introduced a U-statistics-based framework for clustering high-dimensional data, and [25] applied U-statistics techniques to derive the asymptotic properties of estimators based on pairwise differences in a two-way fixed effects model for dyadic interactions. Recent developments include dimension-agnostic inference methods by [26] and the examination of asymmetric U-statistics for pattern matching in random strings by [27]. Additional applications involve network analysis using quadruplet U-statistics [28], empirical risk minimization [29], and methods incorporating left truncation and right censoring [30]. U-statistics also play an important role in random graph theory, particularly in subgraph counting [31], and continue to have broad applicability across diverse fields. The extension of the above exploration to conditional setting is practically useful and technically more challenging.

We now introduce the estimators proposed by [32], which extend the U-statistics concept to conditional expectations in a regression setting. Consider a sequence $\{(\mathbf{X}_i, \mathbf{Y}_i)\}_{i \in \mathbb{N}^*}$ of independent and identically distributed (i.i.d.) random variables, where $\mathbf{X}_i \in \mathbb{R}^d$ denotes the predictor variables and $\mathbf{Y}_i \in \mathcal{Y}$ denotes the response variables, with $\mathcal{Y}$ being a Polish space (a complete separable metric space). Here, $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$ denotes the set of positive integers. Let $\varphi: \mathcal{Y}^k \to \mathbb{R}$ be a measurable function serving as the kernel in this context. Our goal is to estimate the conditional expectation (regression function) $r^{(k)}(\varphi, \tilde{\mathbf{t}})$ at a point $\tilde{\mathbf{t}} = (t_1, \dots, t_k) \in \mathbb{R}^{dk}$, defined as

$$ r^{(k)}(\varphi, \tilde{\mathbf{t}}) = \mathbb{E} [\varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) | (\mathbf{X}_1, \dots, \mathbf{X}_k) = \tilde{\mathbf{t}}], $$

provided it exists, i.e., $\mathbb{E} [\|\varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k)\|] < \infty$. To estimate $r^{(k)}(\varphi, \tilde{\mathbf{t}})$, [32] employed a kernel-based approach. Let $K: \mathbb{R}^d \to \mathbb{R}$ be a kernel function whose support is contained within $[-B, B]^d$ for some $B > 0$, satisfying

$$ \sup_{\mathbf{x} \in \mathbb{R}^d} |K(\mathbf{x})| =: \kappa < \infty \quad \text{and} \quad \int_{\mathbb{R}^d} K(\mathbf{x}) \, d\mathbf{x} = 1. $$

Stute’s class of estimators for $r^{(k)}(\varphi, \cdot)$, known as conditional U-statistics, is defined for each $\tilde{\mathbf{t}} \in \mathbb{R}^{dk}$ as

$$ r_n^{(k)}(\varphi, \tilde{\mathbf{t}}; h_K) = \frac{\sum_{(i_1, \dots, i_k) \in I_n^k} \varphi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \prod_{j=1}^k K\left(\frac{\mathbf{t}_j - \mathbf{X}_{i_j}}{h_K}\right)}{\sum_{(i_1, \dots, i_k) \in I_n^k} \prod_{j=1}^k K\left(\frac{\mathbf{t}_j - \mathbf{X}_{i_j}}{h_K}\right)}, $$

where $\{h_K := h_n\}_{n \ge 1}$ is a sequence of positive bandwidth parameters converging to zero such that $nh_K^{dk} \to \infty$ as $n \to \infty$. In the special case $k=1$, the function $r^{(k)}(\varphi, \tilde{\mathbf{t}})$ reduces to the usual regression function $r^{(1)}(\varphi, \mathbf{t}) = \mathbb{E}[\varphi(\mathbf{Y}) | \mathbf{X} = \mathbf{t}]$, and Stute’s estimator simplifies to the well-known Nadaraya–Watson estimator for nonparametric regression [33,34], as detailed in [35–37]. As an alternative to conventional kernel-type estimators, [38] introduced symmetrized closest neighbor conditional U-statistics; see also [39,40]. Recent developments are discussed in [40,41], where challenges related to uniform bandwidth consistency in general settings are examined.

This work is devoted to the analysis of random observations taking values on the unit hypersphere, defined by $S^d = \{\mathbf{x} \in \mathbb{R}^{d+1} : \|\mathbf{x}\| = 1\}$, for dimensions $d \in \mathbb{N}$. Hyperspherical data naturally arise in astronomy, meteorology, and zoology disciplines. For example, in astronomy, celestial coordinates are often represented on the celestial sphere. In meteorology, wind directions are modeled using circular or spherical data. In zoology, the migratory paths of animals can be projected onto the Earth’s surface, approximated by a sphere. However, these data are frequently contaminated with measurement errors due to limitations in the instruments used for data collection. Accurately recording phenomena such as wind direction, earthquake epicenters, migratory paths of birds or whales, thunderstorms, typhoons, and the dispersion of air pollutants (e.g., PM2.5) on Earth — or sunspot locations on the Sun — is inherently challenging. This difficulty is exacerbated by factors such as the rapid movement of the objects being measured, their substantial distances from measuring devices, and the presence of intervening substances
(e.g., atmospheric particles) between the objects and the instruments, which can further compromise data accuracy [42,43]. Furthermore, in certain applications, Euclidean data are normalized to have a unit norm to ensure that subsequent analyses emphasize relative directions rather than absolute magnitudes. When the original Euclidean data contain measurement errors, this normalization process inevitably propagates the errors onto the hypersphere, complicating statistical inference. Analyzing such *hyperspherical data* presents significant challenges due to the lack of a vector space structure on $S^d$, which complicates the application of conventional statistical techniques that rely on linearity and Euclidean geometry. These data types encompass circular data when $d = 1$, spherical data when $d = 2$, and extend to higher-dimensional analogues for $d > 2$; see [44,45] for comprehensive treatments. Contemporary research on hyperspherical data — assuming the absence of measurement errors — has explored a wide array of topics, including density estimation [46,47], regression analysis [48,49], and statistical hypothesis testing [50,51]. Notably, [48] investigated regression scenarios where both the response and predictor variables are hyperspherical. In their study, the response is assumed to be symmetrically distributed around the product of an unknown orthogonal matrix and the predictor. However, it is important to note that their analysis did not account for the presence of measurement errors. For a comprehensive review of methodologies and challenges in hyperspherical data analysis, see [52]. Despite the critical importance of accurately analyzing hyperspherical data in the presence of measurement errors, existing literature has predominantly focused on specific low-dimensional cases. For instance, [53] studied deconvolution density estimation on the circle $S^1$, pertinent to circular data. [54,55] extended similar deconvolution methods to the sphere $S^2$, addressing applications relevant to spherical data. Additionally, [56] examined deconvolution density estimation on general compact and connected Lie groups, including $S^1$ and $S^3$, focusing on the convergence rates of their estimators. More recently, [57] explored deconvolution regression for predictors defined on general compact and connected Lie groups, offering one of the rare instances where regression analysis is conducted for manifold-valued predictors in the presence of measurement errors. However, it is crucial to note that the unit hypersphere $S^d$ for dimensions $d = 2$ and $d \ge 4$ does not possess a Lie group structure. In the presence of measurement errors, [58] introduced a novel independence test grounded in mutual information, where the requisite conditional density functions are estimated through a newly designed deconvolution-based double kernel procedure. Meanwhile, for high-dimensional partially linear varying coefficient models with covariates in the nonparametric part subject to additive errors, [59] derived the asymptotic distributions of both a corrected empirical log-likelihood ratio function and the maximum empirical likelihood estimator for the regression parameter. We highlight that the methodologies developed in the aforementioned studies are not directly applicable to these higher-dimensional hyperspheres.

When analyzing a dataset to address a specific question, practitioners must carefully evaluate the assumptions underlying their statistical models to determine the reliability of their results. Experienced analysts understand that these assumptions rarely hold exactly in practice and develop an intuition — sometimes supported by formal tests — for assessing the impact of deviations. One of the most prevalent and challenging discrepancies between theoretical models and real-world data is the presence of missing values, which are ubiquitous in modern statistical applications. A fundamental concept in missing-data research is *missing at random* (MAR) [60]. For example, missingness can occur when data are aggregated from sources that measure different variables, such as in healthcare, where patient records may vary across hospitals. Other common causes include individuals omitting sensitive information, data loss due to uncontrollable factors, intermittent surveys, participant dropout [61], sensor failures, data censoring, and privacy concerns. A substantial body of research has addressed this problem within various branches of statistics; for detailed discussions and in-depth perspectives, the reader is referred to [62,63]. According to [64], missing data mechanisms fall into three categories: *missing completely at random* (MCAR), *missing at random* (MAR), and *not missing at random* (NMAR). MCAR represents the most straightforward but least realistic scenario, where missingness is entirely independent of observed and unobserved data. MAR, a more practical assumption, occurs when missingness depends on observed variables but not the missing values. In contrast, NMAR arises when missingness is directly related to the missing values, making it particularly difficult to address. While NMAR may seem intuitively more natural, empirical studies [65] suggest that methods assuming MAR often yield more accurate predictions. Furthermore, in many practical situations, the MAR assumption is frequently justified [64]. Despite extensive research, recent studies [66,67] argue that missing data mechanisms remain poorly understood and are often inaccurately articulated. While the standard formulation [64, p. 12] appears intuitive, it can be misleading in its details; more precise alternatives exist [68], though they often lack heuristic appeal. For further discussion, see [69]. Given these complexities, extending deconvolution techniques and regression methodologies to accommodate hyperspherical data missing at random in higher dimensions is crucial for advancing statistical analysis in this domain. Addressing this theoretical gap is expected to enhance the understanding of manifold statistical methodologies while enabling meaningful advancements in applied scientific disciplines that require rigorous analysis of hyperspherical data, particularly under measurement error conditions. Developing rigorous statistical frameworks that effectively capture the inherent complexities of high-dimensional hyperspherical data is essential for producing accurate and interpretable results.

The primary aim of this study is to develop a comprehensive framework for conditional *U*-statistics of arbitrary order, focusing on missing data observed on a unit hypersphere, often affected by measurement errors. We propose a novel deconvolution technique tailored for conditional *U*-statistics under the missing at-random mechanism and provide the first examination of its convergence rates and asymptotic distribution. The proposed approach establishes a unified and rigorous theoretical framework that is the basis of broad asymptotic properties across a wide range of model structures. In addition, we derive asymptotic confidence intervals from the estimator's distribution, offering practical tools for inference in the presence of measurement errors in the missing data setting. The intricacies lie in the details, and extending previous results from the Nadaraya–Watson estimator to conditional *U*-statistics involves significantly different proofs and requires substantial novel techniques. To the best of our knowledge, our results in this setting are the first established in full generality.
The paper is organized as follows: Section 2 establishes the theoretical framework by introducing key concepts and notation used throughout, complemented by a comprehensive review of harmonic analysis on the hypersphere. In Section 3, we detail the methodological framework and articulate the key assumptions that underpin the subsequent asymptotic analyses. The main theoretical contributions, encompassing convergence rates and asymptotic distributions, are thoroughly explored in Sections 4 and 5, respectively. Section 6 discusses practical applications of our findings, such as the construction of confidence intervals and an examination of the limiting behavior of the conditional Kendall tau's estimator. Finally, Section 7 provides a concise summary of the key findings and their broader implications. Most technical arguments and detailed proofs are relegated to the Appendix to maintain the flow of the main exposition.

## 2. Preliminaries

Let $\{X_i, Y_i, \delta_i\} : i \ge 1\}$ denote a sequence of observations subject to incompleteness, where each triplet $(X_i, Y_i, \delta_i)$ is identically distributed with the generic random vector $(X, Y, \delta)$. In this setting, $X$ is a random vector taking values on the $d$-dimensional unit sphere $S^d$, and $Y$ is a random vector in $\mathbb{R}^d$. The observations $X_i$ for $1 \le i \le n$ are fully observed, whereas the observations $Y_i$ are only partially available. The observability of $Y_i$ is determined by an indicator variable $\delta_i$, where $\delta_i = 1$ if $Y_i$ is observed, and $\delta_i = 0$ otherwise. The indicator variable $\delta$ is a Bernoulli random variable whose conditional probability depends only on $X$, given by

$$ \mathbb{P}(\delta = 1 | \mathbf{Y}, \mathbf{X} = \mathbf{t}) = \mathbb{P}(\delta = 1 | \mathbf{X} = \mathbf{t}) = p(\mathbf{t}), $$

where $p(\mathbf{t})$ is a function representing the probability of observing $\mathbf{Y}$ given $\mathbf{X} = \mathbf{t}$. To facilitate a clear and rigorous exposition of the results, we start by introducing the essential notation, basic definitions, and illustrative examples. This preliminary framework is designed to establish a precise conceptual and methodological bases upon which the subsequent analysis and discussion are systematically developed. Let us recall the setting from [70]. A function $f: \mathbb{R}^{d+1} \to \mathbb{C}$ is called a harmonic homogeneous polynomial of degree $\Re \in \mathbb{N}_0 := \{0\} \cup \mathbb{N}$ in $d+1$ variables if it can be written as

$$ f(t_1, \dots, t_{d+1}) = \sum_{\alpha \in \mathbb{N}_0^{d+1}, |\alpha| = \Re} c_{\alpha} \prod_{i=1}^{d+1} t_i^{\alpha_i}, $$

where $c_\alpha \in \mathbb{C}$, $\alpha = (\alpha_1, \dots, \alpha_{d+1})$, and $|\alpha| = \sum_{i=1}^{d+1} \alpha_i$. Moreover, $f(\cdot)$ satisfies the Laplace equation

$$ \sum_{i=1}^{d+1} \frac{\partial^2 f}{\partial t_i^2} = 0. $$

The restriction of $f(\cdot)$ to the unit sphere $S^d$, denoted by $f|_{S^d}: S^d \to \mathbb{C}$, is called a spherical harmonic of order $\Re$ in $d+1$ variables. The set of all such functions forms a vector space denoted by $\mathfrak{B}^{\Re}(S^d)$, referred to as the spherical harmonic space of order $\Re$ in $d+1$ variables. It is known that the dimension of this space is

$$ N(d, \mathfrak{R}) = \frac{(2\mathfrak{R} + d - 1)(\mathfrak{R} + d - 2)!}{\mathfrak{R}!(d - 1)!}. $$

Specifically, for $\Re \in \mathbb{N}$, we have $N(1, \mathfrak{R}) = 2$ and $N(2, \mathfrak{R}) = 2\mathfrak{R} + 1$, and for any $d$, $N(d, 0) = 1$. Moreover, the union of these spaces $\{\mathfrak{B}^{\Re}(S^d) : \Re \in \mathbb{N}_0\}$ forms a dense subspace of the Hilbert space $L^2(S^d, \nu)$, where $\nu(\cdot)$ is the scaled spherical measure on $S^d$ defined by

$$ \nu(A) = \frac{\text{area}(S^d)}{\text{Leb}(B(0, 1))} \cdot \text{Leb}(\{ta : t \in [0, 1], a \in A\}), $$

for any Borel subset $A \subseteq S^d$. Here, $\text{area}(S^d)$ denotes the surface area of $S^d$, $\text{Leb}$ is the Lebesgue measure on $\mathbb{R}^{d+1}$, and $B(0, 1)$ is the closed ball of radius one centered at the origin. In particular, $\nu(S^d) = \text{area}(S^d)$. The space $L^2(S^d, \nu)$ is a separable Hilbert space with inner product

$$ \langle f, g \rangle = \int_{S^d} f(x) \overline{g(x)} dv(x), $$

where $\overline{g(x)}$ denotes the complex conjugate of $g(x)$. If $\{B_q^{\Re} : 1 \le q \le N(d, \mathfrak{R})\}$ is an orthonormal basis of $\mathfrak{B}^{\Re}(S^d)$, then the collection $\{B_q^{\Re} : \Re \in \mathbb{N}_0, 1 \le q \le N(d, \mathfrak{R})\}$ forms an orthonormal basis of $L^2(S^d, \nu)$. Hereafter, $\mathfrak{R}$ appearing in $B_q^{\Re}$ and other superscripts serve as an index for notational simplicity rather than as an exponent. It is important to note that the constant function $B_1^0 = (\nu(S^d))^{-1/2}$ constitutes the orthonormal basis of $\mathfrak{B}^0(S^d)$ since every spherical harmonic of order zero is constant. Below, we present examples of orthonormal bases for $\mathfrak{B}^{\Re}(S^d)$ with $\Re \in \mathbb{N}$.

### Examples

* **Case $d=1$:**

Every point $x \in S^1$ on the unit circle can be expressed using an angle $\varphi_x \in [0, 2\pi)$ as:

$$ x = \begin{pmatrix} \cos \varphi_x \\ \sin \varphi_x \end{pmatrix}. $$
We define the functions $B_q^{\mathfrak{R}}: \mathbb{S}^1 \to \mathbb{C}$ for $\mathfrak{R} \in \mathbb{N}$ by:

$$B_1^{\mathfrak{R}}(\mathbf{x}) = \frac{1}{\sqrt{\pi}} \cos(\mathfrak{R}\varphi_x), \quad B_2^{\mathfrak{R}}(\mathbf{x}) = \frac{1}{\sqrt{\pi}} \sin(\mathfrak{R}\varphi_x).$$

Then, the set $\{B_q^{\mathfrak{R}}: 1 \le q \le 2\}$ forms an orthonormal basis of the space $\mathfrak{B}^{\mathfrak{R}}(\mathbb{S}^1)$ of spherical harmonics of degree $\mathfrak{R}$ on $\mathbb{S}^1$. For further details, see Chapter 2.2 in [71].

### Case $d = 2$:

Each point $\mathbf{x} \in \mathbb{S}^2$ on the unit sphere can be represented using spherical coordinates $(\varphi_x, \theta_x)$, where $\varphi_x \in [0, 2\pi)$ and $\theta_x \in [0, \pi)$, as:

$$\mathbf{x} = \begin{pmatrix} \cos \varphi_x \sin \theta_x \\ \sin \varphi_x \sin \theta_x \\ \cos \theta_x \end{pmatrix}.$$

We define the functions $B_q^{\mathfrak{R}}: \mathbb{S}^2 \to \mathbb{C}$ for $\mathfrak{R} \in \mathbb{N}$ and $1 \le q \le 2\mathfrak{R} + 1$ by:

$$B_q^{\mathfrak{R}}(\mathbf{x}) = \sqrt{\frac{2\mathfrak{R} + 1}{4\pi}} e^{i(q - 2\mathfrak{R} - 1)\varphi_x} d_{q, 2\mathfrak{R} + 1}^{\mathfrak{R}}(\theta_x),$$

where $d_{qr}^{\mathfrak{R}}(\theta) \in \mathbb{R}$ for $1 \le q, r \le 2\mathfrak{R} + 1$ and $\theta \in [0, \pi)$ is defined by:

$$d_{qr}^{\mathfrak{R}}(\theta) = c_{qr}^{\mathfrak{R}} \sum_{m=\max\{0, r-q\}}^{\min\{2\mathfrak{R}+1-q, r-1\}} \frac{(-1)^{m+q-r}(\cos(\theta/2))^{2\mathfrak{R}-2} m+r-q(\sin(\theta/2))^2 m+q-r}{(2\mathfrak{R}+1-q-m)!(r-1-m)!(m+q-r)!m!},$$

with constants:

$$c_{qr}^{\mathfrak{R}} = \sqrt{(2\mathfrak{R} + 1 - q)!(q - 1)!(2\mathfrak{R} + 1 - r)!(r - 1)!}.$$

The set $\{B_q^{\mathfrak{R}}: 1 \le q \le 2\mathfrak{R} + 1\}$ forms an orthonormal basis of $\mathfrak{B}^{\mathfrak{R}}(\mathbb{S}^2)$. For more details, refer to Theorem 2.1.1 in [72], Chapter 12.9 in [73], and Chapter 3.9 in [74].

### Case $d \ge 3$:

For dimensions $d \ge 3$, an orthonormal basis of $\mathfrak{B}^{\mathfrak{R}}(\mathbb{S}^d)$ can be constructed recursively using orthonormal bases of $\mathfrak{B}^j(\mathbb{S}^{d-1})$ for $0 \le j \le \mathfrak{R}$.

First, we define the Legendre polynomial $P_{\mathfrak{R},d+1}: [-1, 1] \to \mathbb{R}$ of degree $\mathfrak{R} \in \mathbb{N}_0$ in $d+1$ variables by:

$$P_{\mathfrak{R},d+1}(t) = \mathfrak{R}! \Gamma\left(\frac{d}{2}\right) \sum_{m=0}^{[\mathfrak{R}/2]} \frac{(-1)^m (1-t^2)^m t^{\mathfrak{R}-2m}}{4^m m! (\mathfrak{R}-2m)! \Gamma\left(m+\frac{d}{2}\right)}.$$

Next, we define the normalized associated Legendre function $\tilde{P}_{\mathfrak{R},d+1,j}: [-1, 1] \to \mathbb{R}$ for $0 \le j \le \mathfrak{R}$ by:

$$\tilde{P}_{\mathfrak{R},d+1,j}(t) = \frac{\sqrt{(2\mathfrak{R} + d - 1)(\mathfrak{R} + d + j - 2)! (1 - t^2)^{j/2}}}{2^{(d-1)/2+j} \sqrt{(\mathfrak{R} - j)!} \Gamma\left(j + \frac{d}{2}\right)} P_{\mathfrak{R}-j, d+1+2j}(t).$$

Let $\{B_r^j: 1 \le r \le N(j, d-1)\}$ be an orthonormal basis of $\mathfrak{B}^j(\mathbb{S}^{d-1})$ for $0 \le j \le \mathfrak{R}$. Each point $\mathbf{x} \in \mathbb{S}^d$ can be written using angular coordinates $(\varphi_x, \theta_{1,x}, \dots, \theta_{d-1,x})$, where $\varphi_x \in [0, 2\pi)$ and $\theta_{m,x} \in [0, \pi)$ for $1 \le m \le d-1$, as:

$$\mathbf{x} = \left( \cos \varphi_x \cdot \prod_{m=1}^{d-1} \sin \theta_{m,x} \cdot \sin \varphi_x \cdot \prod_{m=1}^{d-1} \sin \theta_{m,x} \cdot \cos \theta_{1,x} \cdot \prod_{m=2}^{d-1} \sin \theta_{m,x} \cdot \cos \theta_{2,x} \cdot \prod_{m=3}^{d-1} \sin \theta_{m,x} \dots \cos \theta_{d-1,x} \right)^{\top} \quad (1)$$

We define the functions $B_{r,j}^{\mathfrak{R}}: \mathbb{S}^d \to \mathbb{C}$ for $\mathfrak{R} \in \mathbb{N}$ by:

$$B_{r,j}^{\mathfrak{R}}(\mathbf{x}) = \tilde{P}_{\mathfrak{R},d+1,j} (\cos \theta_{d-1,x}) B_r^j (\mathbf{s}(\varphi_x, \theta_{1,x}, \dots, \theta_{d-2,x})),$$

where $\mathbf{s}(\varphi_x, \theta_{1,x}, \dots, \theta_{d-2,x})$ is the point on $\mathbb{S}^{d-1}$ defined similarly to (1) but with $d-1$ replaced by $d-2$. Then, the set $\{B_{r,j}^{\mathfrak{R}}: 1 \le r \le N(j, d-1), 0 \le j \le \mathfrak{R}\}$ forms an orthonormal basis of $\mathfrak{B}^{\mathfrak{R}}(\mathbb{S}^d)$. For further details, see Chapter 2.11 in [71].

In this paper, we define the following notation for $\mathbb{V}$, which may refer to either $\mathbb{X}$ or $\mathbb{Z}$:

$$\mathbb{V} = (\mathbf{V}_1, \dots, \mathbf{V}_k) \in \mathbb{S}^{d_k}, \quad \mathbb{V}_i = (\mathbf{V}_{i1}, \dots, \mathbf{V}_{i\ell}) \in \mathbb{S}^{d_\ell}, \quad \mathbf{i} = (i_1, \dots, i_\ell) \in \mathcal{I}(\ell, n),$$

$$\mathbb{Y} = (\mathbf{Y}_1, \dots, \mathbf{Y}_k) \in \mathbb{R}^{d_k}, \quad \mathbb{Y}_i = (\mathbf{Y}_{i1}, \dots, \mathbf{Y}_{i\ell}) \in \mathbb{R}^{d_\ell}, \quad \mathbf{i} = (i_1, \dots, i_\ell) \in \mathcal{I}(\ell, n).$$

Throughout, $\mathbb{C}$ denotes positive constants that may change value between occurrences. Additionally, we use the following asymptotic notations for two positive sequences $a_n$ and $b_n$: $a_n \ge b_n$ means that there exists a constant $c$ such that $a_n \ge c \cdot b_n$ for all $n$, while, $a_n \asymp b_n$ indicates that both $a_n \ge b_n$ and $b_n \ge a_n$.
### 3. Estimation framework and approach

This section outlines the methodological framework we use to estimate regression functions when unobservable errors are present. Specifically, our data includes unmeasured rotational noise, represented by special orthogonal matrices SO(d + 1). We start by defining the space of all real, special orthogonal matrices, SO(d + 1), as matrices of size (d + 1) × (d + 1). The vector X is not directly observable in our setting. Instead, our observed data is given by Z = UX, where U represents an unobservable measurement error taken from SO(d + 1), and UX denotes the product of matrix U with vector X.

The main goal of this work is to estimate the conditional expectation, also referred to as the regression function, based on independent and identically distributed (i.i.d.) data pairs {($Y_i$, $Z_i$) : 1 ≤ i ≤ n}. Specifically, we aim to estimate the following expression:

$$r^{(k)}(\varphi, \tilde{\varphi}) = \mathbb{E}[\varphi(Y_1, \dots, Y_k) | (Z_1, \dots, Z_k) = \tilde{\varphi}], \quad \text{for } \tilde{\varphi} \in \mathbb{S}^{dk},$$

where $\mathbb{E}(|\varphi(Y_1, \dots, Y_k)|) < \infty$ ensures the existence of this conditional expectation. In this context, Y represents the response variable in $\mathbb{R}^d$, and the function $\varphi \cdot \mathbb{R}^{dm} \to \mathbb{R}$ is assumed to be measurable. Additionally, we assume that the unobserved measurement errors represented by matrices $U_i$ are independent of the true vectors $X_i$.

#### 3.1. Convolution property and Fourier decompositions

An essential component of our approach is effectively using a convolution property to address unobserved rotational noise. We define the convolution of two functions—g, belonging to $L^2(\text{SO}(d+1), \mu)$, and $f(\cdot)$, belonging to $L^2(\mathbb{S}^d, \nu)$—as follows:

$$(g * f)(\mathbf{t}) = \int_{\text{SO}(d+1)} g(\mathbf{u}) f(\mathbf{u}^{-1}\mathbf{t}) d\mu(\mathbf{u}),$$

where $\mathbf{u}^{-1}$ is the inverse of the rotation matrix $\mathbf{u}$, and $\mathbf{u}^{-1}\mathbf{t}$ denotes the action of this inverse on the point $\mathbf{t}$ via matrix multiplication. The measure $\mu$ is the normalized Haar measure on the rotation group SO(d + 1), ensuring integration over all possible rotations. Next, we introduce the hyperspherical Fourier transform for functions defined on the sphere $\mathbb{S}^d$. For a given degree $\mathfrak{R} \in \mathbb{N}_0$ and order $1 \le q \le N(d, \mathfrak{R})$, where $N(d, \mathfrak{R})$ denotes the number of spherical harmonics at degree $\mathfrak{R}$, the transform is defined by:

$$\Theta_q^{\mathfrak{R}}(f) = \int_{\mathbb{S}^d} f(\mathbf{t}) \overline{\mathcal{B}_q^{\mathfrak{R}}(\mathbf{t})} d\nu(\mathbf{t}),$$

where $\overline{\mathcal{B}}_q^{\mathfrak{R}}$ are the spherical harmonic basis functions of degree $\mathfrak{R}$, $\overline{\mathcal{B}}_q^{\mathfrak{R}}$ denotes their complex conjugate, and $\nu(\cdot)$ is the standard measure on the sphere. This transform generalizes the classical Euclidean Fourier transform to spherical domains. We extend these concepts to define the Fourier transform in the context of rotations. Specifically, for a function $g$ defined on the rotation group SO(d + 1), its Fourier transform at degree $\mathfrak{R}$ is given by:

$$\tilde{\Theta}_{qr}^{\mathfrak{R}}(g) = \int_{\text{SO}(d+1)} g(\mathbf{u}) D_{qr}^{\mathfrak{R}}(\mathbf{u}) d\mu(\mathbf{u}),$$

where $D_{qr}^{\mathfrak{R}}(\mathbf{u})$ are the matrix elements defined as:

$$D_{qr}^{\mathfrak{R}}(\mathbf{u}) = \overline{\langle \mathcal{B}_q^{\mathfrak{R}}(\cdot; \mathbf{u}), \mathcal{B}_r^{\mathfrak{R}} \rangle}_{L^2} = \int_{\mathbb{S}^d} \overline{\mathcal{B}_q^{\mathfrak{R}}(\mathbf{u} \cdot \mathbf{t}) \mathcal{B}_r^{\mathfrak{R}}(\mathbf{t})} d\nu(\mathbf{t}).$$

Here, $\langle \cdot, \cdot \rangle_{L^2}$ denotes the standard inner product in $L^2(\mathbb{S}^d, \nu)$, and $\mathcal{B}_q^{\mathfrak{R}}(\cdot; \mathbf{u})$ represents the spherical harmonic $\mathcal{B}_q^{\mathfrak{R}}$ rotated by $\mathbf{u}$. The matrix elements $D_{qr}^{\mathfrak{R}}(\mathbf{u})$ quantify how the rotated spherical harmonic $\mathcal{B}_q^{\mathfrak{R}}(\mathbf{u} \cdot \mathbf{t})$ projects onto the original basis function $\mathcal{B}_r^{\mathfrak{R}}(\mathbf{t})$. Since rotating a spherical harmonic of degree $\mathfrak{R}$ results in another spherical harmonic of the same degree, the rotated function remains within the space $\mathcal{B}_r^{\mathfrak{R}}(\mathbb{S}^d)$ spanned by these harmonics. Therefore, it can be expressed as a linear combination of the basis elements:

$$\mathcal{B}_q^{\mathfrak{R}}(\mathbf{u} \cdot \mathbf{t}) = \sum_r \langle \mathcal{B}_q^{\mathfrak{R}}(\cdot; \mathbf{u}), \mathcal{B}_r^{\mathfrak{R}} \rangle_{L^2} \mathcal{B}_r^{\mathfrak{R}}(\mathbf{t}).$$

This expansion shows that the matrix elements $D_{qr}^{\mathfrak{R}}(\mathbf{u})$ serve as the coefficients in expressing the rotated basis functions in terms of the original spherical harmonics. They effectively represent the overlap — or inner product — between the rotated basis function $\mathcal{B}_q^{\mathfrak{R}}(\mathbf{u} \cdot \mathbf{t})$ and the standard basis function $\mathcal{B}_r^{\mathfrak{R}}(\mathbf{t})$. These coefficients are crucial for analyzing functions on the rotation group, as they encode how rotations affect the spherical harmonic components of functions defined on $\mathbb{S}^d$.

#### 3.2. Density deconvolution with measurement error

Let us denote the density of the random rotation U concerning the Haar measure $\mu$ as $f_U(\cdot)$. We assume that $f_U(\cdot)$ is square-integrable, which implies that it resides in the space $L^2(\text{SO}(d+1), \mu)$. Consequently, the density $f_Z(\cdot)$ of $Z = UX \in \mathbb{S}^d$ with respect to $\nu$ exists and is given by $f_Z(\cdot) = f_U * f_X(\cdot)$. For the expression

$$\tilde{f}_Z(\tilde{\mathbf{z}}) := \prod_{j=1}^{k} f_Z(z_j) = \prod_{j=1}^{k} (f_U * f_X)(z_j),$$
we obtain the following deconvolution property:

$$ \phi_q^{\mathcal{R}}(\tilde{f}_{\mathbb{Z}(\tilde{z})}) = \prod_{j=1}^{k} \sum_r \tilde{\Theta}_{qr}^{\mathcal{R}}(f_U) \Theta_r^{\mathcal{R}}(f_X)(z_j). $$

Let $\Theta^{\mathcal{R}}(f)$ denote the $N(d, \mathfrak{R})$-vector whose $q$th element is $\Theta_q^{\mathcal{R}}(f)$, and let $\tilde{\Theta}^{\mathcal{R}}(g)$ be the $N(d, \mathfrak{R}) \times N(d, \mathfrak{R})$ matrix with $(q, r)$th element $\tilde{\Theta}_{qr}^{\mathcal{R}}(g)$, we can rewrite this as:

$$ \Theta^{\mathcal{R}}(f_{\mathbb{Z}}) = \tilde{\Theta}^{\mathcal{R}}(f_U) \Theta^{\mathcal{R}}(f_X). $$

In this study, we assume that the matrix $\tilde{\Theta}^{\mathcal{R}}(f_U)$ is invertible, enabling the deconvolution to be expressed as:

$$ \Theta^{\mathcal{R}}(\tilde{f}_{\mathbb{X}(\tilde{z})}) = \prod_{j=1}^{k} \tilde{\Theta}_{q}^{\mathcal{R}}(f_U)^{-1} \Theta^{\mathcal{R}}(f_{\mathbb{Z}})(z_j). \quad (2) $$

The invertibility of $\tilde{\Theta}^{\mathcal{R}}(f_U)$ is well-established within the deconvolution density estimation literature for $\mathbb{S}^1$ and $\mathbb{S}^2$ (see, for example, [54,75,76]). A variety of distributions on $SO(d + 1)$, such as the Laplace, Gaussian, and von Mises-Fisher distributions, satisfy this invertibility requirement, with further examples provided in [77–81]. In deconvolution on $\mathbb{S}^2$, it is common to assume that $f_U$ is known, which naturally ensures that $\tilde{\Theta}^{\mathcal{R}}(f_U)$ is fully determined. This assumption is also widely adopted in Euclidean measurement error settings (e.g., [82–85]). Following this convention, we assume $f_U$ is known to lay the groundwork for addressing our new problem. Starting with known measurement error distributions is conventional, as these problems are inherently complex.

We now introduce the product density deconvolution estimators. Since the set $\{B_q^{\mathcal{R}} : \mathfrak{R} \in \mathbb{N}_0, 1 \le q \le N(d, \mathfrak{R})\}$ forms an orthonormal basis for $L^2(\mathbb{S}^d, \nu; \mathbb{C})$, we have

$$ \tilde{f}_{\mathbb{X}}(\tilde{t}) = \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d, \mathfrak{R})} \Theta_q^{\mathcal{R}}(f_X(t_j)) B_q^{\mathcal{R}}(t_j), \quad (3) $$

in the $L^2$ sense. The series in (3) is known as the Fourier-Laplace series of $\tilde{f}_{\mathbb{X}}(\cdot)$. This series converges pointwise under certain smoothness conditions on $\tilde{f}_{\mathbb{X}}(\cdot)$, which will be detailed in the following section. From (2) and (3), we can deduce that

$$ \tilde{f}_{\mathbb{X}}(\tilde{t}) = \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d, \mathfrak{R})} \left( \sum_{r=1}^{N(d, \mathfrak{R})} (\tilde{\Theta}^{\mathcal{R}}(f_U(t_j)))_{q\mathfrak{R}}^{-1} \Theta_r^{\mathcal{R}}(f_{\mathbb{Z}}(t_j)) \right) B_q^{\mathcal{R}}(t_j), \quad (4) $$

where $(\tilde{\Theta}^{\mathcal{R}}(f_U))_{q\mathfrak{R}}^{-1}$ is the $(q, r)$th element of $(\tilde{\Theta}^{\mathcal{R}}(f_U))^{-1}$. We truncate the infinite sum $\sum_{\mathfrak{R}=0}^{\infty}$ to a finite sum $\sum_{\mathfrak{R}=0}^{[T_n]}$, where $0 < T_n < \infty$ is a tuning parameter that diverges as $n \to \infty$, and $[T_n]$ represents the greatest integer less than or equal to $T_n$. Using this truncation, we define

$$ \hat{f}_n^{(k)}(\tilde{t}) = \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \mathrm{Re} \left( \tilde{K}_{T_n} (\tilde{t}, \mathbf{Z}_{l_1}, \dots, \mathbf{Z}_{l_k}) \right), \quad (5) $$

where $\tilde{K}_{T_n}(\cdot, \cdot) : \mathbb{S}^{dk} \times \mathbb{S}^{dk} \to \mathbb{C}$ is defined by

$$ \tilde{K}_{T_n} (\tilde{t}, \tilde{z}) = \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d, \mathfrak{R})} \left( \sum_{r=1}^{N(d, \mathfrak{R})} (\tilde{\Theta}^{\mathcal{R}}(f_U(t_j)))_{q\mathfrak{R}}^{-1} \overline{B_r^{\mathcal{R}}(z_j)} \right) B_q^{\mathcal{R}}(t_j), \quad (6) $$

where $\mathrm{Re}(\tilde{K}_{T_n}(\tilde{t}, \mathbb{Z}))$ is the real part of $\tilde{K}_{T_n}(\tilde{t}, \mathbb{Z})$. It is natural to consider only the real part, as $\tilde{f}_{\mathbb{X}}(\cdot)$ is real-valued. The following proposition demonstrates that $\hat{f}_n^{(k)}(\cdot)$ is a valid density estimator, as it satisfies the requirement of integrating to one.

**Proposition 3.1.** $\int_{\mathbb{S}^d} K_{T_n}(\mathbf{t}, \mathbf{z}) d\nu(\mathbf{t}) = 1$ for any $\mathbf{z} \in \mathbb{S}^d$, so that $\int_{\mathbb{S}^d} \mathrm{Re}(K_{T_n}(\mathbf{t}, \mathbf{z})) d\nu(\mathbf{t}) = 1$ for any $\mathbf{z} \in \mathbb{S}^d$ and

$$ \int_{\mathbb{S}^d k} \hat{f}_n^{(k)}(\tilde{t}) d\nu(\mathbf{t}_1) \cdots d\nu(\mathbf{t}_k) = 1. $$

### 3.3. Conditional U-statistics with measurement error

We now present our deconvolution conditional U-statistics for the regression function $r^{(k)}(\varphi, \cdot)$. The proposed estimator is defined as follows:

$$ \hat{\tilde{r}}_n^{(k)}(\varphi, T_n, \tilde{t}) = \frac{\sum_{(i_1, \dots, i_k) \in I_n^k} \prod_{j=1}^{k} \delta_{ij} \varphi(Y_{i_1}, \dots, Y_{i_k}) \mathrm{Re}(\tilde{K}_{T_n}(\tilde{t}, Z_{i_1}, \dots, Z_{i_k}))}{\sum_{(i_1, \dots, i_k) \in I_n^k} \prod_{j=1}^{k} \delta_{ij} \mathrm{Re}(\tilde{K}_{T_n}(\tilde{t}, Z_{i_1}, \dots, Z_{i_k}))} \quad (7) $$
Consider the U-kernel

$$G_{\varphi, T_n, \tilde{\mathbf{t}}, \tilde{\mathbf{z}}, \tilde{\mathbf{y}}, \tilde{\mathbf{g}}} := \tilde{\mathbf{g}} \cdot \varphi(\tilde{\mathbf{y}}) \cdot \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \tilde{\mathbf{z}}), \tilde{\mathbf{t}}, \tilde{\mathbf{z}} \in \mathbb{S}^{dk}, \tilde{\mathbf{y}} \in \mathbb{R}^{dk}.$$

For the observed sample $(Z_1, Y_1), \dots, (Z_n, Y_n)$, we define the corresponding U-statistics as

$$\begin{aligned} \tilde{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) &= \tilde{U}_n^{(k)}(G_{\varphi, T_n, \tilde{\mathbf{t}}}) = \frac{(n-k)!}{n!} \sum_{i \in I_n^k} G_{\varphi, T_n, \tilde{\mathbf{t}}} (\mathbb{Z}_i, \mathbf{Y}_i, \tilde{\mathbf{g}}_i) \\ &= \frac{(n-k)!}{n!} \sum_{\tilde{\mathbf{i}} \in I_n^k} \prod_{j=1}^k \delta_{i_j} \varphi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \tilde{K}_{T_n}(\tilde{\mathbf{i}}, \mathbf{Z}_{i_1}, \dots, \mathbf{Z}_{i_k}). \end{aligned} \quad (8)$$

In contrast to the asymptotic analysis of $\hat{\mathbf{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$, our approach requires additional assumptions on the conditional expectation $\mathbb{E}(\text{Re}(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbf{Z}_{i_1}, \dots, \mathbf{Z}_{i_k})) | \mathbf{X}_1, \dots, \mathbf{X}_n)$, due to the added complexity introduced by $\varphi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k})$. To address this issue, we define the function $\tilde{K}_{T_n}^* : \mathbb{S}^{dk} \times \mathbb{S}^{dk} \to \mathbb{C}$ as

$$\tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{X}^*) = \prod_{j=1}^k \sum_{\mathfrak{X}_j=0}^{[T_n]} \sum_q \overline{B_q^{\mathfrak{X}(\mathbf{X}_j)}} B_q^{\mathfrak{X}(\mathbf{t}_j)}.$$

Given Eq. (3) and the fact that $\prod_{j=1}^k \mathbb{E}(\overline{B_q^{\mathfrak{X}(\mathbf{X}_j)}}) = \prod_{j=1}^k \Theta_q^{\mathfrak{X}}(f_X(\mathbf{t}_j))$, this kernel allows us to estimate $\tilde{f}_{\mathfrak{X}}(\cdot)$ when $\{\mathbf{X}_i : 1 \le i \le n\}$ are observed. For instance, $\tilde{f}_{\mathfrak{X}}(\cdot)$ can be estimated via:

$$\hat{f}_n^{**(k)}(\tilde{\mathbf{t}}) = \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \text{Re} \left( \tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{X}_{i_1}, \dots, \mathbf{X}_{i_k}) \right),$$

or, more generally,

$$\hat{f}_n^{**(k)}(\tilde{\mathbf{t}}) = \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{X}_{i_1}, \dots, \mathbf{X}_{i_k}).$$

The latter extends the estimator proposed by [86] to settings with measurement error. Similarly, the regression function $r^{(k)}(\varphi, \tilde{\mathbf{t}})$ admits the estimator:

$$\hat{\mathbf{r}}_n^{**(k)}(\varphi, T_n, \tilde{\mathbf{t}}) = (\hat{f}_n^{**(k)}(\tilde{\mathbf{t}}))^{-1} \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \varphi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) \text{Re} \left( \tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{X}_{i_1}, \dots, \mathbf{X}_{i_k}) \right).$$

The following proposition shows that $\mathbb{E}(\text{Re}(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbf{Z})) | \mathbf{X}) = \text{Re}(\tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{X}))$.

**Proposition 3.2.** $\mathbb{E}(\tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{Z}) | \mathbf{X}) = \tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{X})$ for all $\tilde{\mathbf{t}} \in \mathbb{S}^{dk}$, so that $\mathbb{E}(\text{Re}(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbf{Z})) | \mathbf{X}) = \text{Re}(\tilde{K}_{T_n}^*(\tilde{\mathbf{t}}, \mathbf{X}))$ for all $\tilde{\mathbf{t}} \in \mathbb{S}^{dk}$.

The property mentioned above, which we term the "hyperspherical unbiased scoring" property, implies that

$$\mathbb{E}(\hat{\mathbf{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) | \mathbf{X}_{i_1}, \dots, \mathbf{X}_{i_k}) = \mathbb{E}(\hat{\mathbf{r}}_n^{**(k)}(\varphi, T_n, \tilde{\mathbf{t}}) \hat{f}_n^{**(k)}(\tilde{\mathbf{t}}) | \mathbf{X}_{i_1}, \dots, \mathbf{X}_{i_k}).$$

This identity indicates that $\tilde{K}_{T_n}^*(\cdot)$ effectively removes the impact of measurement error on the bias of the numerator in $\hat{\mathbf{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$. It is noteworthy that [85] introduced an Euclidean version of this property, and [87] utilized it in the context of Euclidean measurement error problems. Such unbiased scoring properties are crucial for regression analysis in the presence of measurement error.

# 4. Asymptotic analysis of estimation properties

This section presents a comprehensive asymptotic analysis of the proposed estimator introduced in (7), emphasizing its convergence properties under minimal regularity conditions. Our primary goal is to develop a rigorous theoretical framework that generalizes existing results and addresses several persistent challenges in the literature. The analysis focuses on two key aspects: optimal $L_2$ error minimization and the influence of measurement error distribution smoothness on estimation accuracy. We begin by presenting the fundamental assumptions that underpin our theoretical investigation.

## 4.1. Assumptions

**Assumptions 4.1.** The function $\varphi(\cdot)$ is such that :

* $\varphi(\cdot)$ is bounded, i.e., for some $0 < \mathcal{M} < \infty$:

$$\sup_{(\mathbf{y}_1, \dots, \mathbf{y}_k) \in \mathbb{R}^{dk}} |\varphi(\mathbf{y}_1, \dots, \mathbf{y}_k)| \le \mathcal{M}; \quad (F.i)$$

or
• $\varphi(\cdot)$ is unbounded and satisfies, for some $p > 0$,

$$ \theta_{\delta} := \sup_{(t_1, \dots, t_k) \in \mathbb{S}^{dk}} \mathbb{E} \left( |\varphi(\mathbb{Y})|^{2+p} \mid \mathcal{X} = (t_1, \dots, t_k) \right) < \infty. \qquad (\text{F).ii}) $$

### Assumptions 4.2. assump1

(A1) For some $\hat{m} > d/4$ with $\hat{m} \in \mathbb{N}$, (i) $\tilde{f}_{\mathcal{X}}(\cdot) \in C^{2\hat{m}}(\mathbb{S}^{dk})$ and (ii) $r^{(k)}(\varphi, \cdot) \in C^{2\hat{m}}(\mathbb{S}^{dk})$;

(A2) $\tilde{f}_{\mathcal{X}}(\cdot)$ is bounded away from zero on $\mathbb{S}^{dk}$.

Condition (A1) imposes a smoothness requirement on the target functions. Under (A1)-(i), Theorem 2 in [88] ensures that the series in (3) converges uniformly absolutely to $\tilde{f}_{\mathcal{X}}(\cdot)$. This implies that the series converges absolutely, and this convergence holds uniformly.

## 4.2. Smoothness of measurement error distribution

The asymptotic behavior of our estimators critically depends on the smoothness of the measurement error distribution. In classical Euclidean measurement error problems, two primary smoothness categories are commonly recognized: ordinary smooth and super smooth (see [84]). Additionally, a third class known as log-super-smooth is considered for deconvolution density estimation on the sphere $\mathbb{S}^2$. We extend these three smoothness categories to the general setting of $\mathbb{S}^d$.

To rigorously establish this framework, we introduce the operator norm for complex matrices, denoted by $\| \cdot \|_{\text{op}}$. For a complex matrix M of size $N(d, \mathfrak{R}) \times N(d, \mathfrak{R})$, the operator norm is defined as

$$ \|\mathbf{M}\|_{\text{op}} = \sup \{ \|\mathbf{M}\mathbf{v}\|_{\mathbb{C}^N(d, \mathfrak{R})} : \mathbf{v} \in \mathbb{C}^{N(d, \mathfrak{R})}, \|\mathbf{v}\|_{\mathbb{C}^N(d, \mathfrak{R})} = 1 \}, $$

where $\| \cdot \|_{\mathbb{C}^N(d, \mathfrak{R})}$ denotes the norm in the complex space $\mathbb{C}^{N(d, \mathfrak{R})}$. Building upon this definition, we formalize the three smoothness categories in the general setting of $\mathbb{S}^d$ as follows:

• Ordinary-smooth scenario of order $\beta \ge 0$:

$$ (i) \left\| \left( \tilde{\Theta}^{\mathfrak{R}}(f_U) \right)^{-1} \right\|_{\text{op}} \le c_1 \cdot \mathfrak{R}^{\beta}; \qquad (S1) $$

$$ (ii) \left\| \left( \tilde{\Theta}^{\mathfrak{R}}(f_U) \right)^{-1} \right\|_{\text{op}} \ge c_2 \cdot \mathfrak{R}^{\beta}, $$

for constants $c_1, c_2 > 0$ and all $\mathfrak{R} \in \mathbb{N}$.

• Super-smooth scenario of order $\beta > 0$:

$$ (i) \left\| \left( \tilde{\Theta}^{\mathfrak{R}}(f_U) \right)^{-1} \right\|_{\text{op}} \le c_1 \cdot \mathfrak{R}^{\alpha} \cdot \exp(\gamma \cdot \mathfrak{R}^{\beta}); \qquad (S2) $$

$$ (ii) \left\| \left( \tilde{\Theta}^{\mathfrak{R}}(f_U) \right)^{-1} \right\|_{\text{op}} \ge c_2 \cdot \mathfrak{R}^{\alpha} \cdot \exp(\gamma \cdot \mathfrak{R}^{\beta}), $$

for constants $c_1, c_2, \gamma > 0$, $\alpha \in \mathbb{R}$, and all $\mathfrak{R} \in \mathbb{N}$.

• Log-super-smooth scenario of order $\beta > 0$:

$$ (i) \left\| \left( \tilde{\Theta}^{\mathfrak{R}}(f_U) \right)^{-1} \right\|_{\text{op}} \le c_1 \cdot \mathfrak{R}^{\alpha} \cdot \exp(\gamma \cdot \mathfrak{R}^{\beta}(\log \mathfrak{R} - \xi_1)); \qquad (S3) $$

$$ (ii) \left\| \left( \tilde{\Theta}^{\mathfrak{R}}(f_U) \right)^{-1} \right\|_{\text{op}} \ge c_2 \cdot \mathfrak{R}^{\alpha} \cdot \exp(\gamma \cdot \mathfrak{R}^{\beta}(\log \mathfrak{R} - \xi_2)), $$

for constants $c_1, c_2, \gamma > 0$, $\alpha, \xi_1, \xi_2 \in \mathbb{R}$, and all $\mathfrak{R} \in \mathbb{N}$.

We now provide several examples of distributions that satisfy the smoothness scenarios (S1)–(S3): First, consider the case without measurement error. In this situation, the distribution of $U$ is such that $\mathbb{P}(U = I_{d+1}) = 1$, resulting in $\tilde{\Theta}^{\mathfrak{R}}(f_U) = I_{N(d, \mathfrak{R})}$, where $I_k$ is the $k \times k$ identity matrix. This satisfies (S1) with $\beta = 0$. Another example is the Laplace distribution on $SO(d+1)$ with parameter $\lambda > 0$, where $\tilde{\Theta}^{\mathfrak{R}}(f_U)$ is given by $(1 + \lambda^2 \cdot \mathfrak{R}(\mathfrak{R} + d - 1))^{-1} I_{N(d, \mathfrak{R})}$. This smooth distribution satisfies (S1) with $\beta = 2$. For $d = 1$, the density of this distribution is

$$ f_U(\mathbf{w}) = \frac{1}{\lambda\pi} \left( \frac{1}{1 - \exp(-2\pi/\lambda)} \exp\left(-\frac{\varphi_{\mathbf{w}}}{\lambda}\right) + \frac{1}{\exp(2\pi/\lambda) - 1} \exp\left(\frac{\varphi_{\mathbf{w}}}{\lambda}\right) \right), $$

where $\varphi_{\mathbf{w}} \in [0, 2\pi)$ is the angle corresponding to $\mathbf{w}$. For $d = 2$, the density is

$$ f_U(\mathbf{w}) = \frac{1}{\lambda^2 \pi \cos(a_\lambda \pi) \sin(r_\mathbf{w}/2)} \cos(a_\lambda (\pi - r_\mathbf{w})) \mathbf{I}(r_\mathbf{w} > 0), $$
where $a_{\lambda} = \sqrt{1/4 - \lambda^{-2}} \in \mathbb{C}$ and $r_w = \arccos\left(\frac{\operatorname{Trace}(w)-1}{2}\right) \in [0, \pi]$ (Theorem 3.5 in [54]). The Rosenthal distribution on SO(3), with parameters $\theta \in (0, \pi]$ and $p > 0$, has

$$ \tilde{\Theta}^{\mathcal{R}}(f_U) = \left( \frac{\sin\left(\frac{(2\mathcal{R}+1)\theta}{2}\right)}{(2\mathcal{R}+1)\sin\left(\frac{\theta}{2}\right)} \right)^p \mathbf{I}_{2\mathcal{R}+1}, $$

(see [89]), which satisfies (S1) with $\beta = p$. An example of a super-smooth distribution is the Gaussian distribution on SO(d + 1) with parameter $\lambda > 0$, where $\tilde{\Theta}^{\mathcal{R}}(f_U)$ is given by

$$ \exp\left(-\frac{\lambda^2 \cdot \mathfrak{R}(\mathfrak{R} + d - 1)}{2}\right) \mathbf{I}_{N(d, \mathfrak{R})}. $$

This satisfies (S2) with $\beta = 2$, $\alpha = 0$, and $\gamma = \lambda^2/2$. For $d = 1$, its density is

$$ f_U(\mathbf{w}) = \frac{1}{\lambda\sqrt{2\pi}} \sum_{s \in \mathbb{Z}} \exp\left(-\frac{(\varphi_w + 2\pi s)^2}{2\lambda^2}\right), $$

and for $d = 2$, its density is

$$ f_U(\mathbf{w}) = \sum_{\mathfrak{R}=0}^{\infty} (2\mathfrak{R} + 1) \exp\left(-\frac{\lambda^2 \cdot \mathfrak{R}(\mathfrak{R} + 1)}{2}\right) \sum_{q=-\mathfrak{R}}^{\mathfrak{R}} D_{qq}^{\mathfrak{R}}(\mathbf{w}). $$

Turning to the log-super-smooth scenario, the von Mises-Fisher distribution on SO(d + 1), with concentration parameter $\lambda > 0$ and mean direction $\mathbf{a} \in \text{SO}(d+1)$, is log-super-smooth (Theorem 3 in [90]; Section 5.3 in [91]). Its density is given by

$$ f_U(\mathbf{w}) = c(\lambda, \mathbf{a})^{-1} \exp(\lambda \cdot \operatorname{Trace}(\mathbf{a}^{-1}\mathbf{w})), $$

where $c(\lambda, \mathbf{a})$ is the normalizing constant. For $d = 1$, this distribution satisfies (S3) with $\beta = 1$, $\alpha = 0$, $\gamma = 1$, $\xi_1 = 1 + \log \lambda$, and $\xi_2 = 1 + \log(2\lambda)$. For $d = 2$, it satisfies (S3) with $\beta = 1$, $\alpha = 4$, $\gamma = 1$, $\xi_1 = 1 + \log \lambda$, and $\xi_2 = 1 + \log(3\lambda)$.

### 4.3. Convergence analysis

In this section, we analyze the uniform consistency of the product density estimator $\hat{f}_n^{(k)}(\tilde{\mathbf{t}})$ introduced in (5), and evaluate the $L^2$ error rates of the conditional U-statistic estimators $\hat{r}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$ defined in (7). To support our analysis, we define $C^s(\mathbb{S}^d)$ as the class of functions on $\mathbb{S}^d$ that are $s$ times continuously differentiable. The asymptotic behavior of $T_n$ is then investigated under the following diverging rates:

(T1) For condition (S1): $n^{-k^2/2}T_n^{k(\beta+d)} = o(1)$.

(T2) For condition (S2): $n^{-k^2/2}T_n^{k(a+d)} \exp(k\gamma T_n^\beta} = o(1)$.

(T3) For condition (S3): $n^{-k^2/2}T_n^{k(a+d)} \exp(k\gamma T_n^\beta(\log T_n - \xi_1)} = o(1)$.

We now establish the uniform consistency of $\hat{f}_n^{(k)}(\tilde{\mathbf{t}})$. This result is crucial not only for deriving the $L^2$ error rates of $\hat{r}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$ but also holds significant importance in its own right.

**Proposition 4.3.** Suppose that the Fourier-Laplace series in (3) converges uniformly to $\hat{f}_{\mathcal{X}}(\cdot)$. Under the conditions (S1)-(i)+(T1), (S2)-(i)+(T2), or (S3)-(i)+(T3), we have:

$$ \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \hat{f}_{\mathcal{X}}(\tilde{\mathbf{t}}) \right| = o_P(1). $$

It is noteworthy that the series in (3) attains $\hat{f}_{\mathcal{X}}(\cdot)$ under condition (A1)-(i), as uniform absolute convergence guarantees uniform convergence. A less stringent but sufficient condition is that $\hat{f}_{\mathcal{X}}(\cdot) \in C^{s,k}(\mathbb{S}^{dk})$ for some $s \ge 0$ and $\kappa \in (0, 1]$, with $s+\kappa > \frac{\hat{m}(d+1)}{2}-1$. Here, $C^{s,k}(\mathbb{S}^{dk})$ denotes the space of functions whose $\text{th}$ order partial derivatives are Hölder continuous with exponent $\kappa$ (see Theorem 2.36 in [71] for the case when $k=1$). We derive the $L^2$ convergence rates with these instrumental results.

**Theorem 4.4.** Assume that condition (A1)-(i) holds. Then, the following results are valid:

1. Under (S1)-(i), (T1), and (F.i), we have

$$ \int_{\mathbb{S}^{dk}} \left| \hat{r}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}}) \right|^2 d\nu(\mathbf{t}_1) \cdots d\nu(\mathbf{t}_k) = O_{\mathbb{P}} \left( T_n^{-4\hat{m}} + n^{-k^2} T_n^{k(2\beta+d)} \right). $$

Moreover, if instead of (F.i), (F.ii) is satisfied, we obtain the same rate:

$$ \int_{\mathbb{S}^{dk}} \left| \hat{r}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}}) \right|^2 d\nu(\mathbf{t}_1) \cdots d\nu(\mathbf{t}_k) = O_{\mathbb{P}} \left( T_n^{-4\hat{m}} + n^{-k^2} T_n^{k(2\beta+d)} \right). $$
## 2. Under (S2)-(i), (T2), and (F.i), we have

$$ \int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_{n}, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) = O_{\mathbb{P}}(T_n^{-4\hat{m}} + n^{-k^2} T_n^{k(2\alpha+d)} \exp(2k\gamma T_n^{\beta})). $$

Again, if instead of (F.i), (F.ii) is satisfied, the same rate holds:

$$ \int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_{n}, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) = O_{\mathbb{P}}(T_n^{-4\hat{m}} + n^{-k^2} T_n^{k(2\alpha+d)} \exp(2k\gamma T_n^{\beta})). $$

## 3. Under (S3)-(i), (T3), and (F.i), we have

$$ \int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_{n}, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) = O_{\mathbb{P}}(T_n^{-4\hat{m}} + n^{-k^2} T_n^{k(2\alpha+d)} \exp(2k\gamma T_n^{\beta}(\log T_n - \xi_1))). $$

If, instead of (F.i), (F.ii) is satisfied, the rate remains the same:

$$ \int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_{n}, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) = O_{\mathbb{P}}(T_n^{-4\hat{m}} + n^{-k^2} T_n^{k(2\alpha+d)} \exp(2k\gamma T_n^{\beta}(\log T_n - \xi_1))). $$

We can show that $\int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_{\mathbb{P}}(\tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k)$ achieves the same rates as specified in Theorem 4.4 under assumption (A1)-(i). The $L^2$ error rates mentioned above converge to zero as $n \to \infty$. The term $T_n^{-4\hat{m}}$ in the rates arises from the bias of the estimators, while the remaining terms originate from stochastic variability. We can optimize each $L^2$ error rate by choosing a specific diverging sequence for $T_n$. In particular, we consider the following rates:

(T1') In the case of (S1): $T_n \asymp n^{k^2/(4\hat{m}+k(4\beta+d))}$.

(T2') In the case of (S2): $T_n = C(\log n)^{1/\beta}$ for $0 < C < \left(\frac{k}{2\gamma}\right)^{1/\beta}$.

(T3') In the case of (S3): $T_n = C \left( \frac{\log n}{\log \log n} \right)^{1/\beta}$ for $0 < C < \left(\frac{k\beta}{2\gamma}\right)^{1/\beta}$.

The rate specified in (T1') is optimal in the sense that it balances the asymptotic bias term $T_n^{-4\hat{m}}$ with the asymptotic variance term $n^{-k^2} T_n^{k(2\alpha+d)}$. However, in cases (T2') and (T3'), no choice of $T_n$ can effectively balance the asymptotic bias and variance. This difficulty arises because $T_n$ also affects the exponent in terms like $\exp(2k\gamma T_n^{\beta})$ or $\exp(2k\gamma T_n^{\beta}(\log T_n - \xi_1))$. The upper bounds of $C$ in (T2') and (T3') represent critical thresholds. Beyond these thresholds, the terms $n^{-k^2} T_n^{k(2\alpha+d)} \exp(2k\gamma T_n^{\beta})$ and $n^{-k^2} T_n^{k(2\alpha+d)} \exp(2k\gamma T_n^{\beta}(\log T_n - \xi_1))$, respectively, diverge to infinity. For values of $C$ below these thresholds, these terms are dominated by $T_n^{-4\hat{m}}$.

**Corollary 4.5.** Assume that condition (A1)-(i) holds. Then:

### 1. Under (S1)-(i), (T1'), and either (F.i) or (F.ii), we have

$$ \int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_{n}, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) = O_{\mathbb{P}}(n^{-4k^2\hat{m}/(4\hat{m}+k(2\beta+d)))}. $$

### 2. Under (S2)-(i), (T2'), and either (F.i) or (F.ii), we have

$$ \int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_{n}, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) = O_{\mathbb{P}}(\log n)^{-4\hat{m}/\beta}. $$

### 3. Under (S3)-(i), (T3'), and either (F.i) or (F.ii), we have

$$ \int_{\mathbb{S}^{dk}} |\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_{n}, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) = O_{\mathbb{P}}\left(\left(\frac{\log n}{\log \log n}\right)^{-4\hat{m}/\beta}\right). $$

## 5. Asymptotic distributions

In this section, we investigate the asymptotic normality of the estimator $\hat{\mathbf{r}}_{n}^{(k)}(\varphi, T_n, \cdot)$. In our recent work [92], we established the asymptotic normality of these deconvolution conditional U-statistic estimators on general compact and connected Lie groups. Specifically, in the case $k = 1$, [93] also derived the asymptotic normality of their deconvolution estimators. However, to the best of our knowledge, no asymptotic distribution has been derived for $\mathbb{S}^{dk}$ when $d = 2$ or $d \ge 4$ in the presence of measurement error for our estimator.

Before presenting our results on the asymptotic distribution, we must recall a fundamental concept in the study of U-statistics, namely Hoeffding's decomposition. For a detailed exposition, we refer to [10]. Using the notation $\mathbb{P}_1 \times \cdots \times \mathbb{P}_k L = \int L d(\mathbb{P}_1 \times \cdots \times \mathbb{P}_k)$, the Hoeffding projections [1] of a function $L: S^k \to \mathbb{R}$ are defined by

$$ \pi_{\ell} L(x_1, \dots, x_{\ell}) := \pi_{\ell,k}^{\mathbb{P}} L(x_1, \dots, x_{\ell}) := (\delta_{x_1} - \mathbb{P}) \times \cdots \times (\delta_{x_{\ell}} - \mathbb{P}) \times \mathbb{P}^{k-\ell} L, $$
for $x_i \in S$ and $0 \le \ell \le k$. Notably, $\pi_0 L = \mathbb{P}^k L$, and for $\ell > 0$, $\pi_\ell L$ becomes a completely degenerate function of $\ell$ variables. For an integrable function $L(\cdot)$, these projections lead to a decomposition of the U-statistic

$$U_n(L) := U_n^{(k)}(L) := U_n^{(k)}(L, \mathbb{P}) := \frac{1}{\binom{n}{k}} \sum_{1 \le l_1 < \dots < l_k \le n} L(X_{l_1}, \dots, X_{l_k}),$$

into a sum of U-statistics of orders $\ell \le k$, which are orthogonal if $\mathbb{P}^k L^2 < \infty$, and whose kernels are completely degenerate. This is known as the Hoeffding decomposition:

$$U_n^{(k)}(L) = \sum_{\ell=0}^{k} \binom{k}{\ell} U_n^{(\ell)}(\pi_\ell L).$$

This decomposition arises naturally from the expansion

$$L(x_1, \dots, x_k) = \delta_{x_1} \times \dots \times \delta_{x_k} L = \left( (\delta_{x_1} - \mathbb{P}) + \mathbb{P} \right) \times \dots \times \left( (\delta_{x_k} - \mathbb{P}) + \mathbb{P} \right) L,$$

which breaks down into terms of the form

$$(\delta_{x_{l_1}} - \mathbb{P}) \times \dots \times (\delta_{x_{l_\ell}} - \mathbb{P}) \times \mathbb{P}^{k-\ell} L.$$

It is straightforward to verify that a symmetric function $L$ is $\mathbb{P}$-degenerate of order $r-1$ if and only if $r = \min\{\ell > 0 : \pi_{\ell,k}^{\mathbb{P}} L \neq 0\}$. Thus, $L(\cdot)$ is degenerate of order $r-1 \ge 0$ if and only if its Hoeffding expansion, except for the constant term, starts at term $r$; that is,

$$U_n(L) - \mathbb{P}^k L = \sum_{\ell=r}^{k} \binom{k}{\ell} U_n^{(\ell)}(\pi_\ell L).$$

Considering i.i.d. samples $(\mathbf{X}_i, \mathbf{Y}_i), i \ge 1$ from $\mathbb{P}$ and assuming $L \in L_2(\mathbb{P}^k)$, this is an orthogonal decomposition, and

$$\mathbb{E}[\pi_{\ell} L(\mathbf{X}_{\ell}, \mathbf{Y}_{\ell}) | (\mathbf{X}_2, \mathbf{Y}_2), \dots, (\mathbf{X}_{\ell}, \mathbf{Y}_{\ell})] = 0, \quad \ell \ge 1,$$

where we denote $\mathbf{X}_\ell = (\mathbf{X}_1, \dots, \mathbf{X}_\ell)$ and $\mathbf{Y}_\ell = (\mathbf{Y}_1, \dots, \mathbf{Y}_\ell)$. Thus, the kernels $\pi_\ell L$ are canonical for $\mathbb{P}$ (also called completely degenerate or completely centered). Moreover, the projections $\pi_\ell, \ell \ge 1$, are nested, meaning $\pi_\ell \circ \pi_\ell = \pi_\ell$ if $\ell \le l$, and

$$\mathbb{E}[(\pi_{\ell} L)^2(\mathbf{X}_{\ell}, \mathbf{Y}_{\ell})] \le \mathbb{E}[(L - \mathbb{E}L)^2(\mathbf{X}, \mathbf{Y})] \le \mathbb{E}L^2(\mathbf{X}, \mathbf{Y}).$$

Using the decomposition from [1], we obtain

$$\begin{aligned} \tilde{\mu}_n(\varphi, T_n, \tilde{\mathbf{t}}) := \tilde{U}_n(\varphi, T_n, \tilde{\mathbf{t}}) - \theta_n &= \sum_{\ell=1}^{k} \binom{k}{\ell} \tilde{U}_n^{(\ell)}(\pi_{\ell} G_{\varphi, T_n, \tilde{\mathbf{t}}}) \\ &= k \tilde{U}_n^{(1)}(\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}}) + \sum_{\ell=2}^{k} \binom{k}{\ell} \tilde{U}_n^{(\ell)}(\pi_{\ell} G_{\varphi, T_n, \tilde{\mathbf{t}}}), \end{aligned} \quad (9)$$

where $\theta_n$ denotes the expectation of $\tilde{U}_n$, which equals

$$\theta_n = \int_{\mathbb{S}^{dk}} \prod_{j=1}^{k} p(\mathbf{u}_j) \cdot r^{(k)}(\varphi, \mathbf{u}_1, \dots, \mathbf{u}_k) \prod_{j=1}^{k} \mathrm{Re} \left( K_{T_n}(\mathbf{t}_j, \mathbf{u}_j) \right) f_X(\mathbf{u}_j) \, d\mu(\mathbf{u}_j).$$

Based on the definition of the Hoeffding projections and given the i.i.d. sample $(\mathbf{Z}_1, \mathbf{Y}_1), \dots, (\mathbf{Z}_n, \mathbf{Y}_n)$, for all $(\mathbf{z}, \mathbf{y}) \in \mathbb{S}^d \times \mathbb{R}$, we have

$$\begin{aligned} \pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{z}, \mathbf{y}) &= \mathbb{E} \left[ G_{\varphi, T_n, \tilde{\mathbf{t}}} \left( (\mathbf{z}, \mathbf{Z}_2, \dots, \mathbf{Z}_k), (\mathbf{y}, \mathbf{Y}_2, \dots, \mathbf{Y}_k) \right) \right] - \mathbb{E} G_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{z}, \mathbf{Y}) \\ &= \mathbb{E} \left[ G_{\varphi, T_n, \tilde{\mathbf{t}}} (\mathbf{z}, \mathbf{Y}) \mid (\mathbf{Z}_1, \mathbf{Y}_1) = (\mathbf{z}, \mathbf{y}) \right] - \mathbb{E} G_{\varphi, T_n, \tilde{\mathbf{t}}} (\mathbf{z}, \mathbf{Y}). \end{aligned}$$

Let us introduce the function $S_{\varphi, T_n, \tilde{\mathbf{t}}} : \mathbb{S}^d \times \mathbb{R} \to \mathbb{R}$ defined by

$$S_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{z}, \mathbf{y}) = k \cdot \mathbb{E} \left[ G_{\varphi, T_n, \tilde{\mathbf{t}}} (\mathbf{z}, \mathbf{Y}) \mid (\mathbf{Z}_1, \mathbf{Y}_1) = (\mathbf{z}, \mathbf{y}) \right].$$

With this notation, it becomes straightforward to express

$$k \pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}} = S_{\varphi, T_n, \tilde{\mathbf{t}}} - \mathbb{E} S_{\varphi, T_n, \tilde{\mathbf{t}}}.$$

Furthermore, we observe that

$$\begin{aligned} \mathbb{E} [G_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{z}, \mathbf{Y}) | (\mathbf{Z}_1, \mathbf{Y}_1) = (\mathbf{z}, \mathbf{y})] &= p(\mathbf{t}_1) \cdot \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{z}_1) \right) \int_{\mathbb{R}^{dk}} \int_{\mathbb{S}^{d(k-1)}} \prod_{j=2}^{k} p(\mathbf{t}_j) \\ & \times \varphi(\mathbf{y}, \mathbf{y}_2, \dots, \mathbf{y}_k) \prod_{j=2}^{k} \mathrm{Re} \left( K_{T_n}(\mathbf{t}_j, \mathbf{z}_j) \right) f_{XY}(\mathbf{z}_j, \mathbf{y}_j) \, d\mu(\mathbf{z}_j) \, d\mathbf{y}_j \end{aligned}$$
$$= : p(\mathbf{t}_1) \cdot \mathrm{Re} (K_{T_n}(\mathbf{t}_1, \mathbf{z}_1)) \tilde{\phi}_n(\mathbf{y}).$$

Let us define

$$\mathbf{L}_n(\mathbf{t}_1, \mathbf{z}_1, \mathbf{y}) = p(\mathbf{t}_1) \cdot \mathrm{Re} (K_{T_n}(\mathbf{t}_1, \mathbf{z}_1)) \left( \tilde{\phi}_n(\mathbf{y}) - \prod_{j=2}^{k} f_X(\mathbf{t}_j) p(\mathbf{t}_j) \cdot r^{(k)}(\varphi, \tilde{\mathbf{t}}) \right).$$

To achieve our goal, we introduce the following high-level conditions:

(B1) In the case of (S1): There exists a constant $0 \le q \le d$ such that, for each $\mathbf{t}_1 \in \mathbb{S}^d$,

$$\mathbb{E} \left( \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{z}_1) \right) \right)^2 \right) \ge T_n^{2\beta+q};$$

(B2) In the case of (S2): There exists a constant $0 \le q \le d$ such that, for each $\mathbf{t}_1 \in \mathbb{S}^d$ and $\eta \in (0, 1)$,

$$\mathbb{E} \left( \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{z}_1) \right) \right)^2 \right) \ge T_n^{2\alpha+q} \exp \left( 2\gamma(\eta T_n)^\beta \right);$$

(B3) In the case of (S3): There exist constants $0 \le q \le d$ and $\zeta \in \mathbb{R}$ such that, for each $\mathbf{t}_1 \in \mathbb{S}^d$ and $\eta \in (0, 1)$,

$$\mathbb{E} \left( \left( \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{z}_1) \right) \right)^2 \right) \ge T_n^{2\alpha+q} \exp \left( 2\gamma(\eta T_n)^\beta (\log T_n - \zeta) \right).$$

Regarding $T_n$, we consider the following more flexible rates than the specific ones in (T1')-(T3'):

(T1') $T_n \asymp n^p$ for some $0 < p < 1/(2d - q)$, where $q$ is the constant in (B1);

(T2') $T_n = O((\log n)^{1/\beta})$ for $\beta$ in (S2);

(T3') $T_n = O((\log n / \log \log n)^{1/\beta})$ for $\beta$ in (S3);

(T1'') $T_n \asymp n^p$ for some $1/(2\beta + 8\hat{m} + d) < p < 1/(2\beta + 2d)$.

We observe that the range of $p$ in (T1'') is valid because $\hat{m}$ in (A1) satisfies $\hat{m} > d/4$. The upper bound $1/(2\beta + 2d)$ is necessary to ensure (T1). Before presenting the result, let us denote $\Delta_{\mathbb{S}^d}$ as the Laplace-Beltrami operator associated with $\mathbb{S}^d$—a second-order differential operator acting on twice continuously differentiable functions on $\mathbb{S}^d$. We denote by $\Delta_{\mathbb{S}^d}^s$ the composition of $\Delta_{\mathbb{S}^d}$ applied $s$ times. If a function $g: \mathbb{S}^d \to \mathbb{R}$ is $2s$-times continuously differentiable on $\mathbb{S}^d$, then $\Delta_{\mathbb{S}^d}^s(g): \mathbb{S}^d \to \mathbb{R}$ is well defined and continuous on $\mathbb{S}^d$.

**Theorem 5.1.** Assume that the conditions (S1)-(i), (A1), (A2)-(i), (B1) with $q = d$ (B4) and (T1'') hold and that the Fourier-Laplace series of $\Delta_{\mathbb{S}^d}^{\hat{m}}(\tilde{f}_X)$ and $\Delta_{\mathbb{S}^d}^{\hat{m}}(r^{(k)} \cdot \tilde{f}_X)$ converge absolutely on $\mathbb{S}^{dk}$. Then, it holds that for all $\tilde{\mathbf{t}} \in \mathbb{S}^{dk}$,

$$\sqrt{n} \cdot \frac{\hat{\tilde{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}}) - \mathbb{E} (\mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y})) / \prod_{j=1}^k f_X(\mathbf{t}_j) - \theta_n / \hat{\tilde{r}}_n^{(k)}(\tilde{\mathbf{t}})}{\sqrt{\mathrm{Var}(\mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y})) / \prod_{j=1}^k f_X(\mathbf{t}_j)}} \xrightarrow{d} N(0, 1).$$

**Remark 5.2.** Theorem 5.1 can also be extended to a general function $q$ satisfying condition (B1), although this requires more intricate versions of conditions (A1) and (T'''). Regarding the absolute convergence condition in Theorem 5.1, recall that the Fourier-Laplace series of a function $g: \mathbb{S}^d \to \mathbb{R}$ converges absolutely if $g(\cdot)$ is $2s$-times continuously differentiable for $s > d/4$. However, in specific dimensions, much weaker sufficient conditions are adequate. For instance, on $\mathbb{S}^1$, the Fourier-Laplace series of a function converges absolutely if the function is Hölder continuous with an exponent greater than $1/2$ or if it has bounded variation and is Hölder continuous with any positive exponent; see [94].

# 6. Applications

## 6.1. Confidence intervals

In this section, we derive asymptotic confidence intervals for our conditional U-statistic estimator $\hat{\tilde{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$ by using the theoretical results from Theorem 5.1 for the ordinary-smooth case under condition (B1) with $q = d$. To construct these confidence intervals, we need to estimate the unknown bias and variance as outlined in Theorem 5.1. The bias is given by

$$\text{Bias} = \frac{\mathbb{E}(\mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}{\prod_{j=1}^{k} f_X(\mathbf{t}_j)},$$

and the variance is defined as

$$V_n(\mathbf{t}_1, \varphi) = \frac{\sqrt{\mathrm{Var}(\mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))}}{\prod_{j=1}^{k} f_X(\mathbf{t}_j)}.$$
Since we approximate the bias as zero, we focus on estimating the variance. We employ the following estimator for the variance:

$$ \hat{V}_n^2(\mathbf{t}_1, \varphi) = (\hat{f}_n^{(k)}(\tilde{\mathbf{t}}))^{-2} \left[ \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in \mathcal{I}_n^k} (\mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))^2 - \left( \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in \mathcal{I}_n^k} \mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}) \right)^2 \right]. $$

However, because the bias is approximated as zero, the second term becomes negligible, simplifying the variance estimator to

$$ \hat{V}_n^2(\mathbf{t}_1, \varphi) = (\hat{f}_n^{(k)}(\tilde{\mathbf{t}}))^{-2} \frac{(n-k)!}{n!} \sum_{\mathbf{i} \in \mathcal{I}_n^k} (\mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y}))^2. $$

Let $z_{\alpha/2}$ denote the $(1 - \alpha/2)$-quantile of the standard normal distribution for $\alpha \in (0, 1)$.

**Theorem 6.1.** Assume that conditions (S1)–(i), (A1), (A2)–(i), (T1)’, (B4), and (B1) with $q = d$ hold, and that the Fourier–Laplace series of $\Delta_{\mathbb{S}^{dk}}^{\hat{m}}(\tilde{f}_{\mathbb{S}^{dk}})$ and $\Delta_{\mathbb{S}^{dk}}^{\hat{m}}(r^{(k)}\tilde{f}_{\mathbb{S}^{dk}})$ converge absolutely on $\mathbb{S}^{dk}$. Then, for all $\tilde{\mathbf{t}} \in \mathbb{S}^{dk}$, we have

$$ \sqrt{n} \cdot \frac{\hat{\mathring{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})}{\sqrt{V_n(\mathbf{t}_1, \varphi)/\hat{r}_n^{(k)}(\tilde{\mathbf{t}})}} \xrightarrow{d} N(0, 1). $$

Therefore, an asymptotic $(1 - \alpha) \times 100\%$ confidence interval for $r^{(k)}(\varphi, \tilde{\mathbf{t}})$ is given by

$$ \left( \hat{\mathring{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - z_{\alpha/2} \cdot \frac{\sqrt{V_n(\mathbf{t}_1, \varphi)}}{\sqrt{n}}, \quad \hat{\mathring{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) + z_{\alpha/2} \cdot \frac{\sqrt{V_n(\mathbf{t}_1, \varphi)}}{\sqrt{n}} \right). $$

This result provides a practical method for constructing confidence intervals for the estimator $\hat{\mathring{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$.

## 6.2. Kendall tau

To test the independence of one-dimensional random variables $\mathbf{Y}_1$ and $\mathbf{Y}_2$, [95] introduced a method utilizing the U-statistic $K_n$ with the kernel function:

$$ \varphi((\mathbf{s}_1, \mathbf{t}_1), (\mathbf{s}_2, \mathbf{t}_2)) = \mathbb{1}_{\{(\mathbf{s}_2 - \mathbf{s}_1)(\mathbf{t}_2 - \mathbf{t}_1) > 0\}} - \mathbb{1}_{\{(\mathbf{s}_2 - \mathbf{s}_1)(\mathbf{t}_2 - \mathbf{t}_1) \le 0\}}. \quad (10) $$

The rejection region for this test is defined as $\{\sqrt{n}K_n > \gamma\}$. Extending this approach to the multivariate context, we focus on testing the conditional independence of $\xi$ and $\eta$ within the joint distribution $Y = (\xi, \eta)$ given $X$. We propose a method based on the conditional U-statistic:

$$ \hat{\mathring{r}}_n^{(2)}(\varphi, \tilde{\mathbf{t}}; h) = \frac{\sum_{i \neq j}^{n} \varphi(\mathbf{Y}_i, \mathbf{Y}_j) \delta_1 \delta_2 \operatorname{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_i)) \operatorname{Re}(K_{T_n}(\mathbf{t}_2, \mathbf{Z}_j))}{\sum_{i \neq j}^{n} \delta_1 \delta_2 \operatorname{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_i)) \operatorname{Re}(K_{T_n}(\mathbf{t}_2, \mathbf{Z}_j))}, $$

where $\tilde{\mathbf{t}} = (\mathbf{t}_1, \mathbf{t}_2) \in \mathbb{S}^{2d}$, and $\varphi(\cdot)$ is defined by Kendall's kernel as shown in (10). Assuming that $\xi$ and $\eta$ are $k_1$- and $k_2$-dimensional random vectors, respectively, with $k_1 + k_2 = k$, we consider observations $\mathbf{Y}_1, \dots, \mathbf{Y}_n$ of the pair $(\xi, \eta)$. Our goal is to test the hypothesis:

$H_0: \xi$ and $\eta$ are conditionally independent given $X$ vs. $H_a: H_0$ is not true.

Let $\mathbf{a} = (\mathbf{a}_1, \mathbf{a}_2) \in \mathbb{R}^k$ be a unit vector, i.e., $\|\mathbf{a}\| = 1$, where $\mathbf{a}_1 \in \mathbb{R}^{k_1}$ and $\mathbf{a}_2 \in \mathbb{R}^{k_2}$. Denote the distribution functions of $\xi$ and $\eta$ as $F(\cdot)$ and $G(\cdot)$, respectively. We assume that $F^{a_1}(\cdot)$ and $G^{a_2}(\cdot)$ are continuous for any such unit vector, where

$$ F^{a_1}(t) = \mathbb{P}(\mathbf{a}_1^\top \xi < t), \quad G^{a_2}(t) = \mathbb{P}(\mathbf{a}_2^\top \eta < t), $$

with $\mathbf{a}_i^\top$ denoting the transpose of the vector $\mathbf{a}_i$ for $i = 1, 2$. For $n = 2$, let $\mathbf{Y}^{(1)} = (\xi^{(1)}, \eta^{(1)})$ and $\mathbf{Y}^{(2)} = (\xi^{(2)}, \eta^{(2)})$, where $\xi^{(i)} \in \mathbb{R}^{k_1}$ and $\eta^{(i)} \in \mathbb{R}^{k_2}$ for $i = 1, 2$. We define:

$$ \varphi^{a}(\mathbf{Y}^{(1)}, \mathbf{Y}^{(2)}) = \varphi((\mathbf{a}_{1}^{\top} \xi^{(1)}, \mathbf{a}_{2}^{\top} \eta^{(1)}), (\mathbf{a}_{1}^{\top} \xi^{(2)}, \mathbf{a}_{2}^{\top} \eta^{(2)})). $$

We can apply Corollary 4.5 to derive the subsequent results from this framework.

**Corollary 6.2.** Assume that the condition (A1)-(i) holds. Then,

1. Under (S1)-(i), (T1') and (F.i) or (F.ii), it holds that

$$ \int_{\mathbb{S}^{2d}} |\hat{\mathring{r}}_n^{(2)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(2)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1)dv(\mathbf{t}_2) = O_{\mathbb{P}}\left(n^{-16\hat{m}/(4\hat{m}+2(2\beta+d))}\right). $$

2. Under (S2)-(i), (T2') and (F.i) or (F.ii), it holds that

$$ \int_{\mathbb{S}^{2d}} |\hat{\mathring{r}}_n^{(2)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(2)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1)dv(\mathbf{t}_2) = O_{\mathbb{P}}\left(\left(\log n\right)^{-4\hat{m}/\beta}\right). $$
### 3. Under (S3)-(i), (T3'), and (F.i) or (F.ii), it holds that

$$ \int_{S^{2d}} \hat{\mathcal{F}}_n^{(2)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(2)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1)dv(\mathbf{t}_2) = O_P\left( (\log n / \log \log n)^{-4\hat{m}/\beta} \right). $$

### 6.3. Discrimination problems

We now focus on the discrimination problem outlined in Section 3 of [96] and discussed in [32], using consistent notation and assumptions. Let $\varphi(\cdot)$ be a function that takes on a finite set of values $1, 2, \dots, M$. We define the sets

$$ A_j = \{ (\mathbf{y}_1, \dots, \mathbf{y}_k) \mid \varphi(\mathbf{y}_1, \dots, \mathbf{y}_k) = j \}, \quad \text{for } 1 \le j \le M, $$

which partitions the feature space. Predicting the value of $\varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k)$ is thus equivalent to determining which partition $A_j$ the tuple $(\mathbf{Y}_1, \dots, \mathbf{Y}_k)$ belongs to. For any discrimination (classification) rule $g$, we have the inequality

$$ \mathbb{P}(g(\mathbf{X}) = \varphi(\mathbf{Y})) \le \sum_{j=1}^{M} \int_{\tilde{\mathbf{t}}:g(\tilde{\mathbf{t}})=j} \mathfrak{M}^j(\tilde{\mathbf{t}}) d\mathbb{P}(\tilde{\mathbf{t}}), $$

where

$$ \mathfrak{M}^j(\tilde{\mathbf{t}}) = \mathbb{P}(\varphi(\mathbf{Y}) = j \mid \mathbf{X} = \tilde{\mathbf{t}}), \quad \tilde{\mathbf{t}} \in \mathbb{S}^{dk}. $$

Equality holds when the discrimination rule $g$ is defined as

$$ g_0(\tilde{\mathbf{t}}) = \arg \max_{1 \le j \le M} \mathfrak{M}^j(\tilde{\mathbf{t}}), $$

which is known as the Bayes rule. The corresponding probability of error called the Bayes risk, is given by

$$ \mathbf{L}^* = 1 - \mathbb{P}(g_0(\mathbf{X}) = \varphi(\mathbf{Y})) = 1 - \mathbb{E} \left[ \max_{1 \le j \le M} \mathfrak{M}^j(\mathbf{X}) \right]. $$

Each unknown function $\mathfrak{M}^j(\cdot)$ can be consistently estimated using methods discussed earlier. For each $j \in \{1, \dots, M\}$, we define the estimator

$$ \hat{\mathfrak{M}}_n^j(\tilde{\mathbf{t}}) = \frac{\sum_{(i_1, \dots, i_k) \in I(k,n)} \mathbb{1}\{\varphi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_k}) = j\} \prod_{j=1}^k \delta_{ij} \operatorname{Re}(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbf{Z}_{i_1}, \dots, \mathbf{Z}_{i_k}))}{\sum_{(i_1, \dots, i_k) \in I(k,n)} \prod_{j=1}^k \delta_{ij} \operatorname{Re}(\tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbf{Z}_{i_1}, \dots, \mathbf{Z}_{i_k}))}, $$

and we define the corresponding discrimination rule

$$ g_{0,n}(\tilde{\mathbf{t}}) = \arg \max_{1 \le j \le M} \hat{\mathfrak{M}}_n^j(\tilde{\mathbf{t}}). $$

We then define the probability of error for this rule as

$$ \mathbf{L}_n^* = \mathbb{P}(g_{0,n}(\mathbf{X}) \neq \varphi(\mathbf{Y})). $$

The discrimination rule $g_{0,n}(\cdot)$ is asymptotically consistent with the Bayes rule in the sense that

$$ \mathbf{L}_n^* \to \mathbf{L}^* \quad \text{as } n \to \infty. $$

This convergence follows from the inequality

$$ |\mathbf{L}^* - \mathbf{L}_n^*| \le 2 \mathbb{E} \left[ \max_{1 \le j \le M} \left| \hat{\mathfrak{M}}_n^j(\mathbf{X}) - \mathfrak{M}^j(\mathbf{X}) \right| \right], $$

and the fact that $\hat{\mathfrak{M}}_n^j(\mathbf{X})$ converges to $\mathfrak{M}^j(\mathbf{X})$ in probability as $n \to \infty$.

## 7. Concluding remarks

This study presents a comprehensive framework for conditional U-statistics of arbitrary order, focusing on data missing at random and observed on a general unit hypersphere contaminated by measurement errors—a common challenge in modern statistical analysis. We introduce a novel deconvolution method designed explicitly for conditional U-statistics and provide, for the first time, a rigorous analysis of its convergence rates and asymptotic distribution. Our unified approach establishes broad asymptotic properties under general model conditions and derives asymptotic confidence intervals based on the estimator's distribution. Furthermore, we explore practical applications of our results, offering new insights into the Kendall rank correlation coefficient and addressing issues related to statistical discrimination.

Despite these advancements, determining optimal smoothing parameters remains an open problem requiring further investigation. Future research could extend our approach to scenarios involving incomplete data, which present complex mathematical
challenges beyond the current scope of this work. Another promising direction is applying our findings to serially dependent data, particularly under mixing or ergodic conditions. Such extensions could yield valuable insights, especially in the context of change-point detection—a critical area for identifying abrupt shifts in stochastic systems and an unexplored application domain for conditional U-statistics.

Moreover, recent advances in statistics and machine learning have renewed interest in infinite-order U-statistics, particularly for uncertainty quantification in ensemble methods such as Breiman's bagging and random forests. In these contexts, tree-based prediction models can be interpreted as U-statistics where the sub-sample size k scales with the total sample size n. Exploring conditional U-statistics of infinite order constitutes a challenging yet compelling direction for future research, one that requires the development of new methodologies beyond the scope of the present study. Accordingly, we leave this intricate problem for future investigation.

# 8. Mathematical developments

This section is devoted to presenting the proofs of our results. We will consistently use the notation introduced earlier throughout the discussion.

## 8.1. Proof of Proposition 4.3

Let us define

$$ \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) = \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in \mathcal{I}_n^k} \prod_{j} \delta_{ij} \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbf{Z}_{i_1}, \dots, \mathbf{Z}_{i_k}). $$

and set $\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) = \mathrm{Re}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}}))$. Notice that

$$ \begin{aligned} \mathbb{E}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}})) &= \prod_{j=1}^{k} p(\mathbf{t}_j) \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} B_q^{\mathfrak{R}}(\mathbf{t}_j) \sum_{r=1}^{N(d,\mathfrak{R})} (\tilde{\Theta}^{\mathfrak{R}}(f_U(\mathbf{t}_j)))_{qr}^{-1} \Theta_r^{\mathfrak{R}}(f_Z(\mathbf{t}_j)) \\ &= \prod_{j=1}^{k} p(\mathbf{t}_j) \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j), \end{aligned} $$

where the second equality follows from (2). To proceed, we state the following lemma, which will help simplify the expressions.

**Lemma 8.1.** Let $a_i$ and $b_i$, $i = 1, \dots, k$ be real numbers

$$ \prod_{i=1}^{k} a_i - \prod_{i=1}^{k} b_i = \sum_{i=1}^{k} (a_i - b_i) \prod_{j=1}^{l-1} b_j \prod_{\ell=1+l}^{k} a_{\ell}. \quad (11) $$

By applying this lemma, we derive the following

$$ \begin{aligned} \sup_{\tilde{\mathbf{t}} \in S^{dk}} |\tilde{p}(\tilde{\mathbf{t}}) \tilde{f}_X(\tilde{\mathbf{t}}) - \mathbb{E}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}}))| &= \sup_{\tilde{\mathbf{t}} \in S^{dk}} \left| \prod_{j=1}^{k} p(\mathbf{t}_j) \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) - \prod_{j=1}^{k} p(\mathbf{t}_j) \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right| \\ &= \sup_{\tilde{\mathbf{t}} \in S^{dk}} \left| \sum_{j=1}^{k} \left( \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) - \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right) \right. \\ & \qquad \left. \times \prod_{a=1}^{j-1} p(\mathbf{t}_a) \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_a)) B_q^{\mathfrak{R}}(\mathbf{t}_a) \times \prod_{b=1+j}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_b)) B_q^{\mathfrak{R}}(\mathbf{t}_b) \right| \\ &= \sup_{\tilde{\mathbf{t}} \in S^{dk}} \left| \sum_{j=1}^{k} \left( \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right) \right. \\ & \qquad \left. \times \prod_{a=1}^{j-1} p(\mathbf{t}_a) \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_a)) B_q^{\mathfrak{R}}(\mathbf{t}_a) \times \prod_{b=1+j}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \Theta_q^{\mathfrak{R}}(f_X(\mathbf{t}_b)) B_q^{\mathfrak{R}}(\mathbf{t}_b) \right| \\ &= o(1), \end{aligned} $$

where the last equality holds because we assume the Fourier-Laplace series of $f_X(\cdot)$ converges uniformly. Furthermore, we have

$$ \begin{aligned} & \sup_{\tilde{\mathbf{t}} \in S^{dk}} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \mathbb{E}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}}))| \\ & \le \sup_{\tilde{\mathbf{t}} \in S^{dk}} \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \left| \sum_{q=1}^{N(d,\mathfrak{R})} B_q^{\mathfrak{R}}(\mathbf{t}_j) \sum_{r=1}^{N(d,\mathfrak{R})} (\tilde{\Theta}^{\mathfrak{R}}(f_U(\mathbf{t}_j)))_{qr}^{-1} \right| \\ & \qquad \left( \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in \mathcal{I}_n^k} \overline{B_r^{\mathfrak{R}}(\mathbf{Z}_{i_j})} - \Theta_r^{\mathfrak{R}}(f_Z(\mathbf{t}_j)) \right) \end{aligned} $$
$$
\begin{align*}
&= \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \left\| \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \left| \mathbf{B}^{\mathfrak{R}}(\mathbf{t}_j)^{\mathrm{T}} \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right| \left( \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \overline{\mathbf{B}^{\mathfrak{R}}(\mathbf{Z}_{i_j})} - \mathcal{O}^{\mathfrak{R}}(f_{\mathrm{Z}}(\mathbf{t}_j)) \right) \right\| \\
&\le \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \left\| \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \left| \mathbf{B}^{\mathfrak{R}}(\mathbf{t}_j) \right| \right\| \cdot \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \left( \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \overline{\mathbf{B}^{\mathfrak{R}}(\mathbf{Z}_{i_j})} - \mathcal{O}^{\mathfrak{R}}(f_{\mathrm{Z}}(\mathbf{t}_j)) \right) \right\| \\
&\le \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \left\| \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \left| \mathbf{B}^{\mathfrak{R}}(\mathbf{t}_j) \right| \right\| \cdot \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right\|_{\mathrm{op}} \cdot \left\| \left( \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \overline{\mathbf{B}^{\mathfrak{R}}(\mathbf{Z}_{i_j})} - \mathcal{O}^{\mathfrak{R}}(f_{\mathrm{Z}}(\mathbf{t}_j)) \right) \right\| \\
&= \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \sqrt{\frac{N(d, \mathfrak{R})}{v(\mathbb{S}^d)}} \cdot \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right\|_{\mathrm{op}} \cdot \left\| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \overline{\mathbf{B}^{\mathfrak{R}}(\mathbf{Z}_{i_j})} - \mathcal{O}^{\mathfrak{R}}(f_{\mathrm{Z}}(\mathbf{t}_j)) \right\|,
\end{align*}
$$

where the last equality is deduced from the fact

$$
\| \mathbf{B}^{\mathfrak{R}}(\mathbf{t}_j) \|^2 \equiv N(d, \mathfrak{R})/v(\mathbb{S}^d).
$$

Thus, we derive

$$
\begin{align*}
& \mathbb{E} \left( \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \mathbb{E} (\hat{f}_n^{(k)}(\tilde{\mathbf{t}})) \right| \right) \\
& \le \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \sqrt{\frac{N(d, \mathfrak{R})}{v(\mathbb{S}^d)}} \cdot \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right\|_{\mathrm{op}} \cdot \left( \sum_{r=1}^{N(d, \mathfrak{R})} \mathbb{E} \left( \left| \frac{(n-k)!}{n!} \sum_{(i_1, \dots, i_k) \in I_n^k} \overline{\mathbf{B}_r^{\mathfrak{R}}(\mathbf{Z}_{i_j})} - \mathcal{O}_r^{\mathfrak{R}}(f_{\mathrm{Z}}(\mathbf{t}_j)) \right|^2 \right) \right)^{1/2} \\
& \le \frac{(n-k)^{k/2}}{n!^{k/2}} \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \sqrt{\frac{N(d, \mathfrak{R})}{v(\mathbb{S}^d)}} \cdot \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right\|_{\mathrm{op}} \cdot \left( \sum_{r=1}^{N(d, \mathfrak{R})} \mathbb{E} \left( \left| \mathbf{B}_r^{\mathfrak{R}}(\mathbf{Z}) \right|^2 \right) \right)^{1/2} \\
& = (v(\mathbb{S}^d))^{-k} \frac{(n-k)^{k/2}}{n!^{k/2}} \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right\|_{\mathrm{op}}.
\end{align*}
$$

We note that

$$
N(d, \mathfrak{R}) = \left(2 + \frac{d-1}{\mathfrak{R}}\right) \frac{1}{(d-1)!} \frac{\Gamma(\mathfrak{R} + d - 1)}{\Gamma(\mathfrak{R})}.
$$

As established by [97], we have

$$
\frac{\Gamma(\mathfrak{R} + d - 1)}{\Gamma(\mathfrak{R})} = \mathfrak{R}^{d-1} \left( 1 + \frac{(d-1)(d-2)}{2\mathfrak{R}} + O(\mathfrak{R}^{-2}) \right).
$$

Therefore, as $\mathfrak{R} \to \infty$, we find that $N(d, \mathfrak{R}) = O(\mathfrak{R}^{-2})$. Now, let us consider the case (S1)-(i). We have

$$
\prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right\|_{\mathrm{op}} \le \prod_{j=1}^{k} \left( 1 + c_1 \sum_{l=1}^{[T_n]} N(d, \mathfrak{R}) \mathfrak{R}^{\beta} \right) = O \left( T_n^{k(\beta+d)} \right),
$$

where the last equality follows from Faulhaber's formula. By the choice (T1), we get

$$
\sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_X(\tilde{\mathbf{t}})| = \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) + \sqrt{-1} \cdot \operatorname{Im}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}})) - \tilde{f}_X(\tilde{\mathbf{t}})| = o_{\mathbb{P}}(1). \quad (12)
$$

Since the maximum absolute difference between $\hat{f}_n^{(k)}(\tilde{\mathbf{t}})$ and $\tilde{f}_X(\tilde{\mathbf{t}})$ over all $\tilde{\mathbf{t}} \in \mathbb{S}^{dk}$ is less than or equal to that of $\hat{f}_n^{(k)}(\tilde{\mathbf{t}})$, that is,

$$
\sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_X(\tilde{\mathbf{t}})| \le \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_X(\tilde{\mathbf{t}})|,
$$

the desired result immediately follows. For the case labeled (S2)-(i), we obtain

$$
\begin{align*}
\prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\| \left( \tilde{\mathcal{O}}^{\mathfrak{R}}(f_{\mathrm{U}}(\mathbf{t}_j)) \right)^{-1} \right\|_{\mathrm{op}} &\le \prod_{j=1}^{k} \left( 1 + c_1 \sum_{l=1}^{[T_n]} N(d, \mathfrak{R}) \mathfrak{R}^{\alpha} \exp (\gamma \cdot \mathfrak{R}^{\beta}) \right) \\
&= O \left( T_n^{k(\beta+d)} \exp (k \cdot \gamma \cdot T_n^{\beta}) \right).
\end{align*}
$$

We arrive at the desired result by employing the condition (T2). Similarly, applying (S3)-(i) yields the desired result. $\square$

8.2. *Proof of Theorem* **4.4**

Let us define the function $\tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}})$ as follows:

$$
\tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) := r^{(k)}(\varphi, \tilde{\mathbf{t}}) \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}}),
$$
where $r^{(k)}(\varphi, \tilde{\mathbf{t}})$ is the conditional expectation defined earlier, and $\tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}})$ denotes the joint density function of $\mathcal{X} = (\mathbf{X}_1, \dots, \mathbf{X}_k)$ evaluated at $\tilde{\mathbf{t}} = (\mathbf{t}_1, \dots, \mathbf{t}_k)$. Next, we introduce the bias term $\mathbb{B}(\varphi, \tilde{\mathbf{t}})$ by setting

$$ \mathbb{B}(\varphi, \tilde{\mathbf{t}}) := \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) - \mathbb{E}\left[\tilde{U}_{n}^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})\right], $$

where $\tilde{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})$ is the estimator defined in Eq. (8). We observe that

$$ \mathbb{E}\left(\left\|\tilde{U}_{n}^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}}) \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}})\right\|_{2}^{2}\right) = \mathbb{E}\left(\left\|\tilde{U}_{n}^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E}\left(\tilde{U}_{n}^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})\right)\right\|_{2}^{2}\right) + \left\|\mathbb{B}(\varphi, \tilde{\mathbf{t}})\right\|_{2}^{2}. \quad (13) $$

Bounded case

Using condition (F.i), we define

$$ \begin{aligned} \tilde{r}_{B}^{(k)}(\varphi, \tilde{\mathbf{t}}) &:= r_{B}^{(k)}(\varphi, \tilde{\mathbf{t}}) \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}}) \\ &= \mathbb{E}(\Psi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) | (\mathbf{X}_1, \dots, \mathbf{X}_k) = \tilde{\mathbf{t}}) \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}}) \\ &= \mathbb{E}(\varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) \mathbb{1}_{\{\varphi(\mathbf{Y}_1, \dots, \mathbf{Y}_k) \le M\}} | (\mathbf{X}_1, \dots, \mathbf{X}_k) = \tilde{\mathbf{t}}) \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}}). \end{aligned} \quad (14) $$

We begin with the approximation of $\|\mathbb{B}(\varphi, \tilde{\mathbf{t}})\|_2^2$. Notice that

$$ \begin{aligned} \mathbb{E}\left(\prod_{j=1}^{k} \delta_{ij} \Psi(\mathbf{Y}) \prod_{j=1}^{k} \sum_{r=1}^{N(d, \mathcal{R})} \left(\tilde{\Theta}^{\mathcal{R}}(f_{\mathbf{U}}(\mathbf{t}_j))\right)_{qr}^{-1} \overline{B_r^{\mathcal{R}}(\mathcal{Z})}\right) &= \mathbb{E}\left(\mathbb{E}\left[\prod_{j=1}^{k} \delta_{ij} \Psi(\mathbf{Y}) \prod_{j=1}^{k} \sum_{r=1}^{N(d, \mathcal{R})} \left(\tilde{\Theta}^{\mathcal{R}}(f_{\mathbf{U}}(\mathbf{t}_j))\right)_{qr}^{-1} \overline{B_r^{\mathcal{R}}(\mathcal{Z})} \mid \mathcal{X}\right]\right) \\ &= \mathbb{E}\left(\mathbb{E}\left[\prod_{j=1}^{k} \sum_{r=1}^{N(d, \mathcal{R})} \left(\tilde{\Theta}^{\mathcal{R}}(f_{\mathbf{U}}(\mathbf{t}_j))\right)_{qr}^{-1} \overline{B_r^{\mathcal{R}}(\mathcal{Z})} \mid \mathcal{X}\right] \prod_{j=1}^{k} p(\mathbf{t}_j) r_B^{(k)}(\varphi, \tilde{\mathbf{t}})\right) \\ &= \mathbb{E}\left(\prod_{j=1}^{k} p(\mathbf{t}_j) r_B^{(k)}(\varphi, \tilde{\mathbf{t}}) \prod_{j=1}^{k} \overline{B_r^{\mathcal{R}}(\mathcal{X})}\right) \\ &= \Theta_q^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_B^{(k)} \cdot \tilde{f}_{\mathcal{X}}), \end{aligned} \quad (15) $$

where the second equality follows from the underlying assumption $\mathbf{U} \perp \mathbf{X}$. According to (15), we have

$$ \mathbb{E}(\tilde{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}})) = \prod_{j=1}^{k} \sum_{\mathcal{R}=0}^{[T_n]} \sum_{q=1}^{N(d, \mathcal{R})} \Theta_q^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_B^{(k)} \cdot \tilde{f}_{\mathcal{X}}) B_q^{\mathcal{R}}(\mathbf{t}_j). \quad (16) $$

Since the expression is a partial sum of the Fourier-Laplace series for the function $\tilde{\mathbf{p}} \cdot r_B^{(k)} \cdot \tilde{f}_{\mathcal{X}}$ evaluated at $\mathbf{t}_j$, and given that this function is $2\hat{m}$-times continuously differentiable, it follows by applying Eq. (11) that

$$ \begin{aligned} \|\mathbb{B}(\varphi, \tilde{\mathbf{t}})\|_{2}^{2} &= \left\|\tilde{r}_{B}^{(k)}(\varphi, \tilde{\mathbf{t}}) - \mathbb{E}(\tilde{U}_{n}^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}))\right\|_{2}^{2} \\ &= \left\|\prod_{j=1}^{k} \sum_{\mathcal{R}=0}^{\infty} \sum_{q=1}^{N(d, \mathcal{R})} \Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}}) B_{q}^{\mathcal{R}}(\mathbf{t}_{j}) - \prod_{j=1}^{k} \sum_{\mathcal{R}=0}^{[T_n]} \sum_{q=1}^{N(d, \mathcal{R})} \Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}}) B_{q}^{\mathcal{R}}(\mathbf{t}_{j})\right\|_{2}^{2} \\ &= \sum_{j=1}^{k} \sum_{\mathcal{R}>T_n} \sum_{q=1}^{N(d, \mathcal{R})} \left|\Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \times \prod_{a=1}^{j-1} \sum_{q=1}^{N(d, \mathcal{R})} \left|\Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \times \prod_{b=1+j}^{k} \sum_{\mathcal{R}=0}^{\infty} \sum_{q=1}^{N(d, \mathcal{R})} \left|\Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \\ &= \sum_{j=1}^{k} \sum_{\mathcal{R}>T_n} \sum_{q=1}^{N(d, \mathcal{R})} \lambda_{\mathcal{R}}^{-2\hat{m}} \left|(-1)^{\hat{m}} \lambda_{\mathcal{R}}^{\hat{m}} \Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \times \prod_{a=1}^{j-1} \sum_{q=1}^{N(d, \mathcal{R})} \left|\Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \\ &\quad \times \prod_{b=1+j}^{k} \sum_{\mathcal{R}=0}^{\infty} \sum_{q=1}^{N(d, \mathcal{R})} \left|\Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \\ &= \sum_{j=1}^{k} (T_n (T_n + d - 1))^{-2\hat{m}} \sum_{\mathcal{R}>T_n} \sum_{q=1}^{N(d, \mathcal{R})} (T_n (T_n + d - 1))^{2\hat{m}} \lambda_{\mathcal{R}}^{-2\hat{m}} \left|\Theta_{q}^{\mathcal{R}}(\Delta_{\mathcal{S}^{dk}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}}))\right|^{2} \\ &\quad \times \prod_{a=1}^{j-1} \sum_{q=1}^{N(d, \mathcal{R})} \left|\Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \times \prod_{b=1+j}^{k} \sum_{\mathcal{R}=0}^{\infty} \sum_{q=1}^{N(d, \mathcal{R})} \left|\Theta_{q}^{\mathcal{R}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right|^{2} \\ &\le \sum_{j=1}^{k} (T_n (T_n + d - 1))^{-2\hat{m}} \left\|\Delta_{\mathcal{S}^{dk}}^{\hat{m}}(\tilde{\mathbf{p}} \cdot r_{B}^{(k)} \cdot \tilde{f}_{\mathcal{X}})\right\|_{2}^{2}, \end{aligned} \quad (17) $$

where the last inequality follows from the assumption that the Fourier-Laplace series of $\tilde{\mathbf{p}} \cdot r_B^{(k)} \cdot \tilde{f}_{\mathcal{X}}$ converges uniformly, and $-\lambda_{\mathcal{R}} = -\Re(\mathcal{R} + d - 1)$ are the eigenvalues of $\Delta_{\mathcal{S}^{dk}}^{\hat{m}}$. Since $\Delta_{\mathcal{S}^{dk}}^{\hat{m}}(\tilde{\mathbf{p}} \cdot r_B^{(k)} \cdot \tilde{f}_{\mathcal{X}})$ is a continuous function on $\mathcal{S}^{dk}$, it follows that its squared
$L^2$-norm is finite:

$$ \| \Lambda_{\mathbb{S}^{dk}}^{\tilde{m}} (\tilde{p} \cdot r_B^{(k)} \cdot \tilde{f}_{\mathcal{X}}) \|_2^2 < \infty. $$

This implies that

$$ \| \mathbb{B}(\varphi, \tilde{\mathbf{t}}) \|_2^2 = O(T_n^{-4\hat{m}}). $$

Now, we find the rate of

$$ \mathbb{E} \left( \left\| \tilde{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = \int_{\mathbb{S}^{dk}} \mathrm{Var} \left( \tilde{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k). $$

Notice that

$$ \begin{aligned} \mathrm{Var} \left( \tilde{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) &= \frac{(n-k)!}{n!} \mathrm{Var} \left( \prod_{j=1}^{k} \delta_{ij} \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \varphi(\mathbb{Y}) \right) \\ &\le M^2 \frac{(n-k)!}{n!} \mathbb{E} \left( \left| \tilde{K}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \right|^2 \right). \end{aligned} $$

Therefore, we get that

$$ \begin{aligned} \mathbb{E} \left( \int_{\mathbb{S}^{dk}} \left| \tilde{K}_{T_n}(\tilde{\mathbf{t}}, \mathbb{Z}) \right|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) \right) &= \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \mathbb{E} \left( \sum_{q=1}^{[N(d,\mathfrak{R})]} \left| \sum_{r=1}^{[N(d,\mathfrak{R})]} (\tilde{\Theta}^{\mathfrak{R}}(f_{\mathbf{U}}(\mathbf{t}_j)))_{qr} \right|^{-1} \frac{1}{B_r^{\mathfrak{R}}(\mathbb{Z})} \right) \\ &\le \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \left\| (\tilde{\Theta}^{\mathfrak{R}}(f_{\mathbf{U}}(\mathbf{t}_j)))^{-1} \right\|_{\mathrm{op}}^2 \mathbb{E} \left( \left\| \mathbf{B}^{\mathfrak{R}}(\mathbb{Z}) \right\|^2 \right) \\ &= \prod_{j=1}^{k} (\nu(\mathbb{S}^d))^{-1} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\| (\tilde{\Theta}^{\mathfrak{R}}(f_{\mathbf{U}}(\mathbf{t}_j)))^{-1} \right\|_{\mathrm{op}}^2, \end{aligned} $$

where the first equality follows from the orthonormality of {$B_q^{\mathfrak{R}} : 1 \le q \le N(d, \mathfrak{R})$}. In the case of (S1)-(i), we obtain

$$ \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\| (\tilde{\Theta}^{\mathfrak{R}}(f_{\mathbf{U}}(\mathbf{t}_j)))^{-1} \right\|_{\mathrm{op}}^2 \le \prod_{j=1}^{k} \left( 1 + c_1^2 \sum_{l=1}^{[T_n]} N(d, \mathfrak{R}) l^{2\beta} \right) = O(T_n^{k(2\beta+d)}). $$

So, we get

$$ \mathbb{E} \left( \left\| \tilde{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = O \left( T_n^{-4\hat{m}} + \frac{(n-k)!}{n!} T_n^{k(2\beta+d)} \right). $$

In the case of (S2)-(i), we have

$$ \begin{aligned} \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\| (\tilde{\Theta}^{\mathfrak{R}}(f_{\mathbf{U}}(\mathbf{t}_j)))^{-1} \right\|_{\mathrm{op}}^2 &\le \prod_{j=1}^{k} \left( 1 + c_1^2 \sum_{l=1}^{[T_n]} N(d, \mathfrak{R}) l^{2\alpha} \exp(2 \cdot \gamma \cdot T_n^{\beta}) \right) \\ &= O(T_n^{k(2\alpha+d)} \exp(2 \cdot k \cdot \gamma \cdot T_n^{\beta})). \end{aligned} $$

Hence, we obtain

$$ \mathbb{E} \left( \left\| \tilde{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = O \left( T_n^{-4\hat{m}} + \frac{(n-k)!}{n!} T_n^{k(2\alpha+d)} \exp(2 \cdot k \cdot \gamma \cdot T_n^{\beta}) \right). $$

In the case of (S3)-(i), by following the same reasoning, we have

$$ \mathbb{E} \left( \left\| \tilde{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(k)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = O \left( T_n^{-4\hat{m}} + \frac{(n-k)!}{n!} T_n^{k(2\alpha+d)} \exp \left( 2 \cdot k \cdot \gamma \cdot T_n^{\beta} (\log T_n - \xi_1) \right) \right). $$

Now, we note that

$$ \begin{aligned} \inf_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}})| &\ge \inf_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} (\tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}}) - |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}})|) \\ &\ge \inf_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}}) - \sup_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} |\hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}})|. \end{aligned} \quad (18) $$

By invoking **Proposition 4.3** and using the fact that $\inf_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \tilde{f}_{\mathcal{X}}(\tilde{\mathbf{t}}) > 0$, we deduce that there exists a constant $C > 0$ such that

$$ \inf_{\tilde{\mathbf{t}} \in \mathbb{S}^{dk}} \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) > C, $$

with probability approaching one. Therefore, we obtain

$$ \int_{\mathbb{S}^{dk}} |\hat{f}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) $$
$$
\begin{align}
&\le 2 \int_{\mathbb{S}^{dk}} \frac{\left| \hat{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \tilde{\Gamma}_B^{(k)}(\varphi, \tilde{\mathbf{t}}) \right|^2}{\left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) \right|^2} + \frac{r_B^{(k)}(\varphi, \tilde{\mathbf{t}})^2 \left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_X^{(k)}(\tilde{\mathbf{t}}) \right|^2}{\left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) \right|^2} dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) \nonumber \\
&\le C \cdot \left( \left\| \hat{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \tilde{\Gamma}_B^{(k)}(\varphi, \tilde{\mathbf{t}}) \right\|^2 + \left\| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \tilde{f}_X^{(k)}(\tilde{\mathbf{t}}) \right\|^2 \right), \tag{19}
\end{align}
$$

with probability tending to one. Therefore, this completes the proof for the bounded case.

Unbounded case

To prove Theorem 4.4 under the assumption (F.ii), we proceed by truncating the conditional U-statistic. Let $\gamma_n$ be an increasing sequence of positive numbers such that $\gamma_n \to \infty$ as $n \to \infty$. For any function $\varphi$ and any point $\tilde{\mathbf{t}} \in \mathbb{G}^k$, we define

$$
\begin{align*}
G_{\varphi, T_n, \tilde{\mathbf{t}}}(\tilde{\mathbf{z}}, \tilde{\mathbf{y}}) &= G_{\varphi, T_n, \tilde{\mathbf{t}}}(\tilde{\mathbf{z}}, \tilde{\mathbf{y}}) \mathbb{1}_{\{\varphi(\tilde{\mathbf{y}}) \le \gamma_n\}} + G_{\varphi, T_n, \tilde{\mathbf{t}}}(\tilde{\mathbf{z}}, \tilde{\mathbf{y}}) \mathbb{1}_{\{\varphi(\tilde{\mathbf{y}}) > \gamma_n\}} \\
&=: G_{\varphi, T_n, \tilde{\mathbf{t}}}^{(T)}(\tilde{\mathbf{z}}, \tilde{\mathbf{y}}) + G_{\varphi, T_n, \tilde{\mathbf{t}}}^{(R)}(\tilde{\mathbf{z}}, \tilde{\mathbf{y}}),
\end{align*}
$$

which means that we truncate the function $\varphi$ as follows :

$$
\varphi(\tilde{\mathbf{y}}) = \varphi(\tilde{\mathbf{y}}) \mathbb{1}_{\{\varphi(\tilde{\mathbf{y}}) \le \gamma_n\}} + \varphi(\tilde{\mathbf{y}}) \mathbb{1}_{\{\varphi(\tilde{\mathbf{y}}) > \gamma_n\}} = \varphi^{(T)}(\tilde{\mathbf{y}}) + \varphi^{(R)}(\tilde{\mathbf{y}}). \quad (20)
$$

We can also observe that $\tilde{U}_n^{(k)}(G_{\varphi, T_n, \tilde{\mathbf{t}}})$, as described in (8), can be decomposed as follows:

$$
\tilde{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) = \tilde{U}_n^{(k)}(G_{\varphi, T_n, \tilde{\mathbf{t}}}^{(T)}) + \tilde{U}_n^{(k)}(G_{\varphi, T_n, \tilde{\mathbf{t}}}^{(R)}) =: \tilde{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) + \tilde{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}). \quad (21)
$$

We denote the term on the right-hand side as $\tilde{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}})$, referred to as the truncated component, and the second term as $\tilde{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}})$, referred to as the remainder component. Our analysis begins with $\tilde{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}})$.

Truncated part

First, we find the rate of

$$
\mathbb{E} \left( \left\| \tilde{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = \int_{\mathbb{S}^{dk}} \mathrm{Var} \left( \tilde{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k).
$$

By applying condition (F.ii), we observe that

$$
\begin{align*}
\mathrm{Var} \left( \tilde{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) &= \frac{(n-k)!}{n!} \mathrm{Var} \left( \tilde{\mathfrak{g}} \cdot \tilde{\mathcal{K}}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \varphi^{(T)} (\mathbb{Y}) \right) \\
&\leq \frac{(n-k)!}{n!} \mathbb{E} \left( \left| \tilde{\mathfrak{g}} \cdot \tilde{\mathcal{K}}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \varphi^{(T)} (\mathbb{Y}) \right|^2 \right) \\
&\leq \frac{(n-k)!}{n!} \mathbb{E} \left( \mathbb{E} \left[ \left| \tilde{\mathcal{K}}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \right|^2 \mid \mathbb{X} \right] \mathbb{E} \left[ \left| \varphi^{(T)} (\mathbb{Y}) \right|^2 \mid \mathbb{X} \right] \mathbb{E} \left[ \left| \tilde{\mathfrak{g}} \right|^2 \mid \mathbb{X} \right] \right) \\
&\leq \theta_{\delta}^{\frac{2+\delta}{n}} \frac{(n-k)!}{n!} \mathbb{E} \left( \left| \tilde{\mathcal{K}}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \right|^2 \right),
\end{align*}
$$

where the second inequality follows from the underlying assumption U⊥X. In the case of (S1)-(i), we get

$$
\mathbb{E} \left( \left\| \tilde{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = O \left( T_n^{-4\hat{m}} + \frac{(n-k)!}{n!} \cdot T_n^{k(2\beta+d)} \right). \quad (22)
$$

This implies that

$$
\begin{align*}
\left\| \hat{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \hat{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 &\le \left\| \tilde{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(T)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \\
&= O_{\mathbb{P}} \left( T_n^{-4\hat{m}} + \frac{(n-k)!}{n!} \cdot T_n^{k(2\beta+d)} \right),
\end{align*}
$$

where

$$
\hat{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) = \mathrm{Re} \left( \tilde{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) \right).
$$

Remainder part

We now turn our attention to approximating the remainder term $\tilde{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}})$. Following a strategy akin to our treatment of the
truncated component, we begin by establishing the rate of convergence of

$$
\mathbb{E} \left( \left\| \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = \int_{\mathbb{S}^{dk}} \mathrm{Var} \left( \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k).
$$
Notice that, taking into consideration condition (F.ii),

$$
\begin{align*}
\mathrm{Var} \left( \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) &\le \frac{(n-k)!}{n!} \mathbb{E} \left( |\tilde{\mathbf{g}} \cdot \tilde{\mathbf{K}}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z}) \varphi^{(R)} (\mathbb{Y})|^2 \right) \\
&\le \frac{(n-k)!}{n!} \mathbb{E} \left( \mathbb{E} \left[ |\tilde{\mathbf{K}}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z})|^2 \mid \mathbb{X} \right] \mathbb{E} \left[ |\varphi^{(R)} (\mathbb{Y})|^2 \mid \mathbb{X} \right] \mathbb{E} \left[ |\tilde{\mathbf{g}}|^2 \mid \mathbb{X} \right] \right) \\
&\le \theta_{\delta}^{\frac{2+\alpha}{2}} \frac{(n-k)!}{n!} \mathbb{E} \left( |\tilde{\mathbf{K}}_{T_n} (\tilde{\mathbf{t}}, \mathbb{Z})|^2 \right),
\end{align*}
$$

where the second inequality follows from the assumption $\mathbf{U} \perp \mathbf{X}$. In the case of (S1)-(i), we obtain

$$
\mathbb{E} \left( \left\| \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right) = O \left( T_n^{-4\hat{m}} + \frac{(n-k)!}{n!} \cdot T_n^{k(2\hat{\beta}+d)} \right). \quad (23)
$$

Denote $\hat{\mathbb{U}}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}) = \mathrm{Re}(\tilde{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}))$. As a result, we infer that

$$
\begin{align*}
\left\| \hat{\mathbb{U}}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \hat{\mathbb{U}}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 &\le \left\| \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n^{(R)} (\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \\
&= O_{\mathbb{F}} \left( T_n^{-4\hat{m}} + \frac{(n-k)!}{n!} \cdot T_n^{k(2\hat{\beta}+d)} \right).
\end{align*}
$$

We now focus on analyzing the bias term $\|\mathbb{B}(\varphi, \tilde{\mathbf{t}})\|_2^2$. Our approach will closely mirror the methodology employed for the bias in the bounded case. To begin, recall the definition $\tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) := r^{(k)}(\varphi, \tilde{\mathbf{t}}) \tilde{f}_{\mathbb{X}}(\tilde{\mathbf{t}})$. Using this, we can express the bias term as follows

$$
\begin{align*}
\mathbb{E} \left( \tilde{\mathbf{g}} \cdot \varphi (\mathbb{Y}) \prod_{j=1}^{k} \sum_{r=1}^{N(d;\mathbb{R})} \left( \tilde{\Theta}^{\mathbb{R}} (f_{\mathbb{U}}(\mathbf{t}_j)) \right)_{qr}^{-1} \overline{B_r^{\mathbb{R}}(\mathbb{Z})} \right) &= \mathbb{E} \left( \mathbb{E} \left[ \tilde{\mathbf{g}} \cdot \varphi (\mathbb{Y}) \prod_{j=1}^{k} \sum_{r=1}^{N(d;\mathbb{R})} \left( \tilde{\Theta}^{\mathbb{R}} (f_{\mathbb{U}}(\mathbf{t}_j)) \right)_{qr}^{-1} \overline{B_r^{\mathbb{R}}(\mathbb{Z})} \mid \mathbb{X} \right] \right) \\
&= \mathbb{E} \left( \mathbb{E} \left[ \prod_{j=1}^{k} \sum_{r=1}^{N(d;\mathbb{R})} \left( \tilde{\Theta}^{\mathbb{R}} (f_{\mathbb{U}}(\mathbf{t}_j)) \right)_{qr}^{-1} \overline{B_r^{\mathbb{R}}(\mathbb{Z})} \mid \mathbb{X} \right] \tilde{p}(\tilde{\mathbf{t}}) \cdot r^{(k)}(\varphi, \tilde{\mathbf{t}}) \right) \\
&= \mathbb{E} \left( \tilde{p}(\tilde{\mathbf{t}}) \cdot r^{(k)}(\varphi, \tilde{\mathbf{t}}) \prod_{j=1}^{k} \overline{B_r^{\mathbb{R}}(\mathbb{X})} \right) \\
&= \Theta_q^{\mathbb{R}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathbb{X}}).
\end{align*}
$$

According to the last equality, we get that

$$
\mathbb{E} \left( \tilde{U}_n (\varphi, T_n, \tilde{\mathbf{t}}) \right) = \prod_{j=1}^{k} \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d;\mathfrak{R})} \Theta_q^{\mathfrak{R}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}}) B_q^{\mathfrak{R}} (\mathbf{t}_j). \quad (24)
$$

By following the same steps as (17), we readily obtain

$$
\begin{align*}
\| \mathbb{B}(\varphi, \tilde{\mathbf{t}}) \|_2^2 &= \| \tilde{r}^{(k)}(\varphi, \tilde{\mathbf{t}}) - \mathbb{E} \left( \tilde{U}_n (\varphi, T_n, \tilde{\mathbf{t}}) \right) \|_2^2 \\
&= \sum_{j=1}^k \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d;\mathfrak{R})} |\Theta_q^{\mathfrak{R}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}})|^2 \times \prod_{a=1}^{j-1} \sum_{q=1}^{N(d;\mathfrak{R})} |\Theta_q^{\mathfrak{R}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}})|^2 \times \prod_{b=1+j}^k \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d;\mathfrak{R})} |\Theta_q^{\mathfrak{R}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}})|^2 \\
&= \sum_{j=1}^k (T_n (T_n + d - 1))^{-2\hat{m}} \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d;\mathfrak{R})} (T_n (T_n + d - 1))^{2\hat{m}} \lambda_{\mathfrak{R}}^{-2\hat{m}} |\Theta_q^{\mathfrak{R}} (\Delta_{\mathfrak{S}^{dk}}^{\hat{m}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}}))|^2 \\
&\quad \times \prod_{a=1}^{j-1} \sum_{q=1}^{N(d;\mathfrak{R})} |\Theta_q^{\mathfrak{R}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}})|^2 \times \prod_{b=1+j}^k \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d;\mathfrak{R})} |\Theta_q^{\mathfrak{R}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}})|^2 \\
&\le \sum_{j=1}^k (T_n (T_n + d - 1))^{-2\hat{m}} \| \Delta_{\mathfrak{S}^{dk}}^{\hat{m}} (\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}}) \|_2^2. \tag{25}
\end{align*}
$$

Since $\Delta_{\mathfrak{S}^{dk}}^{\hat{m}}(\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}})$ is a continuous function on $\mathfrak{S}^{dk}$, it follows that $\|\Delta_{\mathfrak{S}^{dk}}^{\hat{m}}(\tilde{p} \cdot r^{(k)} \cdot \tilde{f}_{\mathfrak{X}})\|_2^2 < \infty$. Consequently, we can deduce that

$$
\| \mathbb{B}(\varphi, \tilde{\mathbf{t}}) \|_2^2 = O \left( T_n^{-4\hat{m}} \right). \qquad (26)
$$

Consequently, we conclude that

$$
\int_{\mathfrak{S}^{dk}} |\hat{\mathbb{r}}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}})|^2 dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k)
$$
$$
\begin{aligned}
& \le 2 \int_{\mathbb{S}^{dk}} \frac{\left| \hat{U}_n^{(k)}(\varphi, T_n, \tilde{\mathbf{t}}) - \bar{\varphi}^{(k)}(\varphi, \tilde{\mathbf{t}}) \right|^2}{\left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) \right|^2} + \frac{r^{(k)}(\varphi, \tilde{\mathbf{t}})^2 \left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \bar{f}_X(\tilde{\mathbf{t}}) \right|^2}{\left| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) \right|^2} dv(\mathbf{t}_1) \cdots dv(\mathbf{t}_k) \\
& \le C \cdot \left( \left\| \hat{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \hat{U}_n^{(T)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 + \left\| \hat{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}) - \mathbb{E} \left( \hat{U}_n^{(R)}(\varphi, T_n, \tilde{\mathbf{t}}) \right) \right\|_2^2 \right. \\
& \qquad \left. + \left\| \mathbb{E} (\varphi, \tilde{\mathbf{t}}) \right\|_2^2 + \left\| \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - \bar{f}_X(\tilde{\mathbf{t}}) \right\|_2^2 \right).
\end{aligned}
$$

with probability tending to one. By applying (18), the desired result is established for case (S1)-(i). Using a similar approach, we
derive the corresponding results for cases (S2)-(i) and (S3)-(i). This concludes the proof of the theorem. □

8.3. *Proof of Theorem 5.1*

We will start by treating the first term in (9).

Linear term of (9)

We first note that

$$
\begin{align}
& \frac{\tilde{U}_n(\varphi, T_n, \tilde{\mathbf{t}}) - \theta_n}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} + \frac{\theta_n}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} - r^{(k)}(\varphi, \tilde{\mathbf{t}}) \tag{27} \\
&= \frac{k U_n^{(1)}(\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} + \frac{\theta_n}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} - r^{(k)}(\varphi, \tilde{\mathbf{t}}) + o(1) \nonumber \\
&= \frac{kn^{-1}}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} \sum_{i=1}^n \left( p(\mathbf{t}_i) \cdot \operatorname{Re} (K_{T_n}(\mathbf{t}_i, \mathbf{Z}_i) \tilde{\varphi}_n(\mathbf{Y}_i)) - \mathbb{E} \left( p(\mathbf{t}_i) \cdot \operatorname{Re} (K_{T_n}(\mathbf{t}_i, \mathbf{Z}_i) \tilde{\varphi}_n(\mathbf{Y}_i)) \right) \right) + \frac{\theta_n}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} - r^{(k)}(\varphi, \tilde{\mathbf{t}}) + o(1). \nonumber
\end{align}
$$

According to (27), we can see that

$$
\begin{align}
\frac{k U_n^{(1)} (\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} &= \frac{1}{\prod_{j=1}^k p(\mathbf{t}_j) f_X(\mathbf{t}_j)} \cdot kn^{-1} \sum_{i=1}^n \mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) \left( 1 + \frac{\prod_{j=1}^k p(\mathbf{t}_j) f_X(\mathbf{t}_j)}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} - 1 \right) \nonumber \\
&= \frac{1}{\prod_{j=1}^k p(\mathbf{t}_j) f_X(\mathbf{t}_j)} \cdot kn^{-1} \sum_{i=1}^n \mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) \nonumber \\
&\quad + \frac{1}{\prod_{j=1}^k p(\mathbf{t}_j) f_X(\mathbf{t}_j)} \cdot kn^{-1} \sum_{i=1}^n \mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i) \cdot \frac{\prod_{j=1}^k p(\mathbf{t}_j) f_X(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} \nonumber \\
&= \sum_{i=1}^n \mathbf{L}_{n,i}(\mathbf{t}_1) + \sum_{i=1}^n \mathbf{L}_{n,i}(\mathbf{t}_1) \cdot \frac{\prod_{j=1}^k p(\mathbf{t}_j) f_X(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})}, \tag{28}
\end{align}
$$

where $\mathbb{L}_{n,i}(\mathbf{t}_1) = \frac{kn^{-1}}{\prod_{j=1}^k p(\mathbf{t}_j)f_X(\mathbf{t}_j)} \mathbf{L}_n(\mathbf{t}_1, \mathbf{Z}_i, \mathbf{Y}_i)$. Hence, we readily infer that

$$
\begin{align*}
& \frac{k U_n^{(1)} (\pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}}) / \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) - r^{(k)}(\varphi, \tilde{\mathbf{t}}) - \mathbb{E} \left( \sum_{i=1}^n \mathbb{L}_{n,i}(\mathbf{t}_1) \right)}{\sqrt{\operatorname{Var} \left( \sum_{i=1}^n \mathbb{L}_{n,i}(\mathbf{t}_1) \right)}} \\
&= \frac{\sum_{i=1}^n \mathbb{L}_{n,i}(\mathbf{t}_1) - \mathbb{E} \left( \sum_{i=1}^n \mathbb{L}_{n,i}(\mathbf{t}_1) \right)}{\sqrt{\operatorname{Var} \left( \sum_{i=1}^n \mathbb{L}_{n,i}(\mathbf{t}_1) \right)}} + \frac{\sum_{i=1}^n \mathbb{L}_{n,i}(\mathbf{t}_1)}{\sqrt{\operatorname{Var} \left( \sum_{i=1}^n \mathbb{L}_{n,i}(\mathbf{t}_1) \right)}} \cdot \frac{\prod_{j=1}^k p(\mathbf{t}_j) f_X(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})}.
\end{align*}
$$

Thus, it suffices to prove that

$$
\frac{\sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_1) - \mathbb{E}(\sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_1))}{\sqrt{\operatorname{Var}(\sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_1))}} \xrightarrow{d} N(0, 1), \qquad (29)
$$

and that

$$
\frac{\sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_1)}{\sqrt{\operatorname{Var}(\sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_1))}} \cdot \frac{\prod_{j=1}^{k} p(\mathbf{t}_j) f_X(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} = o_p(1). \quad (30)
$$
For (29), we check the Lyapunov condition

$$ \frac{\mathbb{E}\left[\left|\mathbb{L}_{n,1}(\mathbf{t}_1) - \mathbb{E}(\mathbb{L}_{n,1}(\mathbf{t}_1))\right|^{2+\delta}\right]}{n^{\delta/2} (\mathrm{Var}(\mathbb{L}_{n,1}(\mathbf{t}_1)))^{1+\delta/2}} \to 0, $$

for $\delta > 0$ in (F.ii). This is equivalent to checking that

$$ \frac{\mathbb{E}\left[\left|\mathbf{L}_n(\mathbf{t}_1) - \mathbb{E}(\mathbf{L}_n(\mathbf{t}_1))\right|^{2+\delta}\right]}{n^{\delta/2} (\mathrm{Var}(\mathbf{L}_n(\mathbf{t}_1)))^{1+\delta/2}} \to 0. $$

Since

$$ \mathbb{E}\left[\left|\mathbf{L}_n(\mathbf{t}_1) - \mathbb{E}(\mathbf{L}_n(\mathbf{t}_1))\right|^{2+\delta}\right] \le C \cdot \mathbb{E}\left[\left|\mathbf{L}_n(\mathbf{t}_1)\right|^{2+\delta}\right], $$

it suffices to show that

$$ \frac{\mathbb{E}\left(\left|\mathbf{L}_n(\mathbf{t}_1)\right|^{2+\delta}\right)}{n^{\delta/2} (\mathrm{Var}(\mathbf{L}_n(\mathbf{t}_1)))^{1+\delta/2}} \to 0. \quad (31) $$

Also, since

$$ \begin{aligned} \mathbb{E}\left(\left|\mathbf{L}_n(\mathbf{t}_1)\right|^{2+\delta}\right) &\le C \cdot \mathbb{E}\left(\left|p(\mathbf{t}_1) \cdot \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))\right|^{2+\delta}\right), \\ \mathbb{E}\left(\left(\mathbf{L}_n(\mathbf{t}_1)\right)^2\right) &\ge C \cdot \mathbb{E}\left(\left(p(\mathbf{t}_1) \cdot \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))\right)^2\right) \to \infty, \\ \mathbb{E}(\mathbf{L}_n(\mathbf{t}_1)) &= \mathbb{E}\left[p(\mathbf{t}_1) \cdot \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1)) \left(\prod_{j=1}^{k-1} p(\mathbf{Z}_j) \cdot f_X(\mathbf{Z}_j)\right)^{r^{(k)}(\varphi, \tilde{\mathbf{Z}})} - \prod_{j=1}^{k-1} p(\mathbf{t}_j) \cdot f_X(\mathbf{t}_j)\right] \to 0, \end{aligned} $$

it suffices to show that

$$ \frac{\mathbb{E}\left[\left|p(\mathbf{t}_1) \cdot \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))\right|^{2+\delta}\right]}{n^{\delta/2} \left(\mathbb{E}\left(\left(p(\mathbf{t}_1) \cdot \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))\right)^2\right)\right)^{1+\delta/2}} \to 0. \quad (32) $$

We note that

$$ \begin{aligned} \mathbb{E}\left(\left|p(\mathbf{t}_1) \cdot K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1)\right|^{2+\delta}\right) &= |p(\mathbf{t}_1)|^{2+\delta} \int_{\mathbb{S}^d} |K_{T_n}(\mathbf{t}_1, \mathbf{w})|^{2+\delta} f_Z(\mathbf{w}) d\nu(\mathbf{w}) \\ &\le \sup_{\mathbf{t}_1 \in \mathbb{S}^d} f_X(\mathbf{t}_1) \int_{\mathbb{S}^d} |K_{T_n}(\mathbf{t}_1, \mathbf{w})|^{2+\delta} d\nu(\mathbf{w}), \end{aligned} \quad (33) $$

where the inequality follows from the fact that $\sup_{w \in \mathbb{S}^d} f_Z(w) \le \sup_{\mathbf{t}_1 \in \mathbb{S}^d} f_X(\mathbf{t}_1)$. We also note that

$$ \begin{aligned} \left|K_{T_n}(\mathbf{t}_1, \mathbf{w})\right|^{\delta} &\le \left(\sum_{\mathfrak{R}=0}^{[T_n]} \|\mathbf{B}^{\mathfrak{R}}(\mathbf{t}_1)\| \cdot \left\|\left(\tilde{\phi}^{\mathfrak{R}}(f_U(\mathbf{t}_1))\right)^{-1}\right\|_{\mathrm{op}} \cdot \|\mathbf{B}^{\mathfrak{R}}(\mathbf{w})\|\right)^{\delta} \\ &= \left(\nu(\mathbb{S}^d)\right)^{-1} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\|\left(\tilde{\phi}^{\mathfrak{R}}(f_U(\mathbf{t}_1))\right)^{-1}\right\|_{\mathrm{op}}^{\delta}, \end{aligned} \quad (34) $$

and that

$$ \begin{aligned} \int_{\mathbb{S}^d} |K_{T_n}(\mathbf{t}_1, \mathbf{w})|^2 d\nu(\mathbf{w}) &= \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{r=1}^{N(d, \mathfrak{R})} \left| \sum_{q=1}^{N(d, \mathfrak{R})} B_q^{\mathfrak{R}}(\mathbf{t}) (\tilde{\phi}^{\mathfrak{R}}(f_U))_{qr}^{-1} \right|^2 \\ &\le \sum_{\mathfrak{R}=0}^{[T_n]} \|\mathbf{B}^{\mathfrak{R}}(\mathbf{t}_1)\|^2 \cdot \left\|\left(\tilde{\phi}^{\mathfrak{R}}(f_U)\right)^{-1}\right\|_{\mathrm{op}}^T \\ &= (\nu(\mathbb{S}^d))^{-1} \sum_{\mathfrak{R}=0}^{[T_n]} N(d, \mathfrak{R}) \left\|\left(\tilde{\phi}^{\mathfrak{R}}(f_U)\right)^{-1}\right\|_{\mathrm{op}}^2, \end{aligned} \quad (35) $$
where the first equality in (35) follows from the orthonormality of {$B_q^{\mathfrak{R}} : 1 \le q \le N(d, \mathfrak{R})$}. Combining (33), (34) and (35), we get

$$
\mathbb{E} \left[ \left| p(\mathbf{t}_1) \cdot (K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1)) \right|^{2+\delta} \right] \le \begin{cases} C \cdot T_n^{(2+\delta)\beta+(1+\delta)d} & \text{if (S1)-(i) holds,} \\ C \cdot T_n^{(2+\delta)\alpha+(1+\delta)d} \exp\left((2+\delta)\gamma \cdot T_n^{\beta}\right) & \text{if (S2)-(i) holds,} \\ C \cdot T_n^{(2+\delta)\alpha+(1+\delta)d} \exp\left((2+\delta)\gamma \cdot T_n^{\beta}(\log T_n - \xi_1)\right) & \text{if (S3)-(i) holds.} \end{cases} \tag{36}
$$

Since

$$
\mathbb{E} \left[ \left| p(\mathbf{t}_1) \cdot \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1) \right) \right|^{2+\delta} \right] \le \mathbb{E} \left[ \left| p(\mathbf{t}_1) \cdot K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1) \right|^{2+\delta} \right],
$$

it follows that $\mathbb{E}\left[\left|p(\mathbf{t}_1) \cdot \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))\right|^{2+\delta}\right]$ is subject to the same upper bounds as those given in (36). Combining this with the lower bound for $\mathbb{E}\left[\left(p(\mathbf{t}_1) \cdot \mathrm{Re}(K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1))\right)^{2+\delta}\right]$ derived in (B1), we confirm the validity of (32) under condition (S1)-(i). Additionally, leveraging the lower bounds established in (B2) and (B3), and selecting $\eta$ sufficiently close to 1, we ensure that (32) holds for the scenarios described in (S2)-(i) and (S3)-(i). Turning to (30), we observe that by applying (11),

$$
\begin{align*}
& \left| \prod_{j=1}^{k} p(\mathbf{t}_j) \cdot f_X(\mathbf{t}_j) - \mathbb{E}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}})) \right| \\
&= \left| \sum_{j=1}^{k} p(\mathbf{t}_j) \cdot \left( \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right) \right. \\
&\qquad \left. \times \prod_{a=1}^{j-1} p(\mathbf{t}_a) \cdot \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_a)) B_q^{\mathfrak{R}}(\mathbf{t}_a) \times \prod_{b=1+j}^{k} p(\mathbf{t}_b) \cdot \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_b)) B_q^{\mathfrak{R}}(\mathbf{t}_b) \right| \\
&\le \sum_{j=1}^{k} \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right| \\
&\qquad \times \prod_{a=1}^{j-1} \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_a)) B_q^{\mathfrak{R}}(\mathbf{t}_a) \right| \times \prod_{b=1+j}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_b)) B_q^{\mathfrak{R}}(\mathbf{t}_b) \right| \\
&= \sum_{j=1}^{k} \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \lambda_{\mathfrak{R}}^{-\hat{m}} \left| (-1)^{\hat{m}} \lambda_{\mathfrak{R}}^{\hat{m}} \phi^{\mathfrak{R}}(f_X(\mathbf{t}_j)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right| \\
&\qquad \times \prod_{a=1}^{j-1} \sum_{\mathfrak{R}=0}^{[T_n]} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_a)) B_q^{\mathfrak{R}}(\mathbf{t}_a) \right| \times \prod_{b=1+j}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X(\mathbf{t}_b)) B_q^{\mathfrak{R}}(\mathbf{t}_b) \right| \\
&\le \sum_{j=1}^{k} (T_n(T_n+d-1))^{-\hat{m}} \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d,\mathfrak{R})} (T_n(T_n+d-1))^{\hat{m}} \lambda_{\mathfrak{R}}^{-\hat{m}} \left| \phi_q^{\mathfrak{R}}(\Delta_{\mathfrak{S}^d}^{\hat{m}}(f_X)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right| \\
&\qquad \times \prod_{a=1}^{j-1} \sum_{\mathfrak{R}=0}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(\tilde{f}_X) B_q^{\mathfrak{R}}(\mathbf{t}_a) \right| \times \prod_{b=1+j}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X) B_q^{\mathfrak{R}}(\mathbf{t}_b) \right| \\
&\le \sum_{j=1}^{k} (T_n(T_n+d-1))^{-\hat{m}} \sum_{\mathfrak{R}>T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(\Delta_{\mathfrak{S}^d}^{\hat{m}}(f_X)) B_q^{\mathfrak{R}}(\mathbf{t}_j) \right| \\
&\qquad \times \prod_{a=1}^{j-1} \sum_{\mathfrak{R}=0}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X) B_q^{\mathfrak{R}}(\mathbf{t}_a) \right| \times \prod_{b=1+j}^{k} \sum_{\mathfrak{R}=0}^{\infty} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}}(f_X) B_q^{\mathfrak{R}}(\mathbf{t}_b) \right| \\
&= o(T_n^{-2\hat{m}}), \tag{37}
\end{align*}
$$

where the last equality holds due to the absolute convergence of the Fourier–Laplace series for $\Delta_{\mathfrak{S}^d}^{\hat{m}}(f_X)$, as well as the assumption
that the Fourier–Laplace series for $r_B^{(k)} \cdot \tilde{f}_X$ converges uniformly. Consequently, it follows that

$$
\mathbb{E}(\hat{f}_n^{(k)}(\tilde{\mathbf{t}})) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) = O_{\mathbb{P}}(n^{-k^2} \cdot T_n^{k(\beta+d)}),
$$

by arguing as in the proof of Proposition 4.3. This, when combined with (37), implies that

$$
\prod_{j=1}^{k} p(\mathbf{t}_j) \cdot f_X(\mathbf{t}_j) - \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) = o(T_n^{-2\hat{m}}) + O_{\mathbb{P}}(n^{-k^2} \cdot T_n^{k(\beta+d)}). \quad (38)
$$
Hence, it suffices to show that

$$ \frac{\mathbb{E} \left( \sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_i) \right)}{\mathrm{Var} \left( \sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_i) \right)} \cdot \frac{o(T_n^{-2\hat{m}}) + O_{\mathbb{P}}(n^{-k^2} \cdot T_n^{k(\beta+d)})}{\hat{f}_n^{(k)}(\tilde{\mathbf{t}})} = o_{\mathbb{P}}(1). \quad (39) $$

We observe that $(\hat{f}_n^{(k)}(\tilde{\mathbf{t}}))^{-1} = O_{\mathbb{P}}(1)$. This result follows directly from (38) and the assumption (A2)-(i). Furthermore, it is worth noting that

$$ \begin{aligned} & \prod_{j=1}^{k} p(\mathbf{t}_j) \cdot f_X(\mathbf{t}_j) \cdot \left| \mathbb{E} \left( \sum_{i=1}^{n} \mathbb{L}_{n,i} (\mathbf{t}_i) \right) \right| \\ &= \left| \mathbb{E} (k \cdot \mathbf{L}_n (\mathbf{t}_1, \mathbf{Z}_1, \mathbf{Y})) \right| \\ &= \left| \mathbb{E} \left[ k \cdot p(\mathbf{t}_1) \cdot \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1) \left( \tilde{\phi}_n(\mathbf{Y}) - \prod_{j=2}^{k} p(\mathbf{t}_j) \cdot f_X(\mathbf{t}_j) r^{(k)}(\varphi, \tilde{\mathbf{t}}) \right) \right] \right| \\ &= k \left| \mathbb{E} \left[ p(\mathbf{t}_1) \cdot \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1) \tilde{\phi}_n(\mathbf{Y}) \right) \right] - \mathbb{E} \left[ p(\mathbf{t}_1) \cdot \mathrm{Re} \left( K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1) \prod_{j=2}^{k} p(\mathbf{t}_j) \cdot f_X(\mathbf{t}_j) r^{(k)}(\varphi, \tilde{\mathbf{t}}) \right) \right] \right| \\ &\le k \left| p(\mathbf{t}_1) \right| \left| r^{(k)}(\varphi, \mathbf{t}_1) \right| \cdot \sum_{\mathfrak{R} > T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}} (f_X) B_q^{\mathfrak{R}}(\mathbf{t}_1) \right| + \sum_{\mathfrak{R} > T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}} (r^{(k)} \cdot \tilde{p} \cdot \tilde{f}_{\mathfrak{R}}) B_q^{\mathfrak{R}}(\tilde{\mathbf{t}}) \right| \\ &\le k (T_n (T_n + d - 1))^{-\hat{m}} \left( \left| r^{(k)}(\varphi, \mathbf{t}_1) \right| \cdot \sum_{\mathfrak{R} > T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}} \left( \Delta_{\mathfrak{S}d}^{\hat{m}} (f_X) B_q^{\mathfrak{R}}(\mathbf{t}_1) \right) \right| \right. \\ & \qquad \left. + \sum_{\mathfrak{R} > T_n} \sum_{q=1}^{N(d,\mathfrak{R})} \left| \phi_q^{\mathfrak{R}} \left( \Delta_{\mathfrak{S}dk}^{\hat{m}} (r^{(k)} \cdot \tilde{p} \cdot \tilde{f}_{\mathfrak{R}}) B_q^{\mathfrak{R}}(\tilde{\mathbf{t}}) \right) \right| \right) \\ &= o(T_n^{-2\hat{m}}), \end{aligned} \quad (40) $$

where the last equality follows from the absolute convergence of the Fourier-Laplace series of $\Delta_{\mathfrak{S}d}^{\hat{m}}(f_X)$ and $\Delta_{\mathfrak{S}dk}^{\hat{m}}(r^{(k)} \cdot \tilde{f}_{\mathfrak{R}})$. Also, it holds that

$$ \mathrm{Var} \left( \sum_{i=1}^{n} \mathbb{L}_{n,i}(\mathbf{t}_i) \right) \geq C n^{-1} \mathbb{E} \left( \left( p(\mathbf{t}_1) \cdot K_{T_n}(\mathbf{t}_1, \mathbf{Z}_1) \right)^2 \right) \geq C \cdot n^{-1} \cdot p^2(\mathbf{t}_1) \cdot T_n^{2\beta+d}. \quad (41) $$

Thus, (39) follows if

$$ n^{1/2} T_n^{-(\beta+q/2)} T_n^{-4\hat{m}} = O(1) \quad \text{and} \quad T_n^{d-q/2} T_n^{-2\hat{m}} = O(1). \quad (42) $$

The first result in (42) is derived from condition (T1''). The second result in (42) is established using conditions (A1)-(i) and (A1)-(ii), with the requirement that $\hat{m} > (2d - q)/4$. This concludes the proof for the linear term.

**Remark 8.2.** We provide a detailed explanation for why we do not cover the cases (S2)-(i) and (S3)-(i). Under (S2)-(i) combined with (B), we obtain the rate $o(T_n^{-2\hat{m}}) + O_{\mathbb{P}}(n^{-k^2/2} \cdot T_n^{k(\alpha+d)} \exp(k \cdot \gamma \cdot T_n^\beta))$ in (39) and derive the lower bound

$$ C \cdot n^{-k^2} \cdot T_n^{k(2\alpha+d)} \exp(2 \cdot k \cdot \gamma \cdot (\eta \cdot T_n)^\beta) \\ \text{in (41). Therefore, for } \hat{f}_n^{(k)}(\tilde{\mathbf{t}}) \text{ to be consistent and for } (\hat{f}^{(k)} n(\tilde{\mathbf{t}}))^{-1} = O_{\mathbb{P}}(1), \text{ we need} \\ n^{-k^2} \cdot T_n^{k(2\alpha+d)} \exp(k \cdot \gamma \cdot T_n^\beta) = o(1). \quad (43) $$

Additionally, to verify a version of (39), we need

$$ n^{-k^2} \cdot T_n^{-k(\alpha+d/2)} \exp(-k \cdot \gamma \cdot (\eta \cdot T_n^\beta) \cdot T_n^{-2\hat{m}} \cdot (T_n^{-2\hat{m}} + n^{-k^2/2} T_n^{k(\alpha+d)} \exp(k \cdot \gamma \cdot T_n^\beta))) = O(1). \quad (44) $$

For (43), a log-type rate for $T_n$ is required. However, (44) does not hold with any log-type rate for $T_n$. The case (S3)-(i) combined with (B3) has a similar issue.

### The other terms of (9)

Let us now focus on the nonlinear component of the Hoeffding decomposition. The goal, as established in [32], is to demonstrate the following convergence:

$$ (nT_n)^{1/2} \left[ \tilde{\mu}_n(\varphi, T_n, \tilde{\mathbf{t}}) - k \tilde{U}_n^{(1)} \left( \pi_1 G_{\varphi, T_n, \tilde{\mathbf{t}}} \right) \right] \to 0 \quad \text{in } L^2. $$

To obtain this result, it is crucial to apply the variance formula for centered U-statistics, which is given by

$$ \mathbb{V}_n(\varphi, T_n, \tilde{\mathbf{t}}) = \frac{(n-k)!}{n!} \sum_{i \in I_n^k} \frac{\tilde{G}_{\varphi, T_n, \tilde{\mathbf{t}}}(\mathbf{Z}_i, \mathbf{Y}_i)}{N}, $$
where $\tilde{G}_{\phi,T_n,\tilde{\mathbf{i}}}(\mathbf{z},\mathbf{y}) = G_{\phi,T_n,\tilde{\mathbf{i}}}(\mathbf{z},\mathbf{y}) - \mathbb{E}(G_{\phi,T_n,\tilde{\mathbf{i}}}(\mathbf{z},\mathbf{y}))$ and $N = \prod_{j=1}^k \mathbb{E}K_{T_n}(\mathbf{t}_j, \mathbf{Z}_1)$, which is given by

$$ \mathrm{Var}(\mathbb{V}_n) = \left(\frac{(n-k)!}{n!}\right)^2 \sum_{s=1}^{k} \frac{(n-s)!}{(n-2k+s)!} \sum_{|\Omega_1|=s=|\Omega_2|}^{(s)} \mathbf{I}(\Omega_1, \Omega_2), $$

and $\sum^{(s)}$ indicates summation over all positions $\Omega_1$ and $\Omega_2$ of length $l$,

$$ \mathbf{I}(\Omega_1, \Omega_2) = \mathbb{E} \left[ \tilde{G}_{\phi, T_n, \tilde{\mathbf{i}}} \left( (\mathbf{Z}_{l_1}, \mathbf{Y}_{l_1}), \dots, (\mathbf{Z}_{l_k}, \mathbf{Y}_{l_k}) \right) \tilde{G}_{\phi, T_n, \tilde{\mathbf{i}}} \left( (\mathbf{Z}_{j_1}, \mathbf{Y}_{j_1}), \dots, (\mathbf{Z}_{j_k}, \mathbf{Y}_{j_k}) \right) \right], $$

and the $i$'s in position $\Omega_1$ correspond to the $j$'s in position $\Omega_2$. All indices $i$ and $j$ are mutually distinct. When this condition is applied to $\mathbb{V}_n = \tilde{\mu}_n - k\tilde{U}_n^{(1)}(\pi_1 G_{\phi,T_n,\tilde{\mathbf{i}}})$, we obtain

$$ \sum^{(1)} \mathbf{I}(\Omega_1, \Omega_2) = 0. $$

Furthermore, we can assert that

$$ N^{-2}\mathbf{I}(\Omega_1, \Omega_2) = O(T_n^{-s}) \quad \text{for each} \quad 2 \le s \le k. $$

This leads to

$$ \begin{aligned} nT_n \mathrm{Var}(\tilde{\mu}_n - k\tilde{U}_n^{(1)}(\pi_1 G_{\phi,T_n,\tilde{\mathbf{i}}})) &= O \left[ nT_n \sum_{l=2}^{k} \binom{n}{s}^{-1} \binom{k}{s} \binom{n-k}{k-s} T_n^{-s} \right] \\ &= O \left[ \sum_{s=2}^{k} (nT_n)^{1-s} \right] \\ &= O[(nT_n)^{-1}] = o(1). \end{aligned} $$

Hence, the proof is concluded. $\square$

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The authors extend their sincere gratitude to the Editor-in-Chief, the Associate Editor, and the referee for their invaluable feedback and for pointing out a number of oversights in the version initially submitted. Their insightful comments have greatly refined and focused the original work, resulting in a markedly improved presentation.

## Appendix

**Proposition A.1** ([93]). Let $f \in L^2((\mathbb{S}^d, \nu), \mathbb{C})$ and $g \in L^2((SO(d+1), \mu), \mathbb{C})$. Then $\Theta_q^{\mathfrak{R}}(g * f) = \sum_{r=1}^{N(d,\mathfrak{R})} \tilde{\Theta}_{qr}^{\mathfrak{R}}(g) \Theta_q^{\mathfrak{R}}(f)$, for all $\mathfrak{R} \in \mathbb{N}_0$ and $1 \le q \le N(d, \mathfrak{R})$.

**Remark A.2.** Modeling measurement error using matrices from the special orthogonal group $SO(d+1)$ is a natural choice, as these matrices represent rotations that preserve the intrinsic geometry of the sphere $\mathbb{S}^d$. For example, in two dimensions, any matrix in $SO(2)$ can be expressed as:

$$ \begin{pmatrix} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{pmatrix}, $$

where $\varphi \in [0, 2\pi)$ represents the angle of a counter-clockwise rotation on the circle $\mathbb{S}^1$. In three dimensions, matrices in $SO(3)$ can be represented as a product of three rotation matrices: $\mathbf{r}_z(\vartheta_z)\mathbf{r}_y(\vartheta_y)\mathbf{r}_x(\vartheta_x)$, where $\vartheta_x, \vartheta_y, \vartheta_z \in [0, 2\pi)$, and the individual rotation matrices are given by:

$$ \mathbf{r}_z(\vartheta) = \begin{pmatrix} \cos \vartheta & -\sin \vartheta & 0 \\ \sin \vartheta & \cos \vartheta & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \mathbf{r}_y(\vartheta) = \begin{pmatrix} \cos \vartheta & 0 & \sin \vartheta \\ 0 & 1 & 0 \\ -\sin \vartheta & 0 & \cos \vartheta \end{pmatrix}, \quad \mathbf{r}_x(\vartheta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \vartheta & -\sin \vartheta \\ 0 & \sin \vartheta & \cos \vartheta \end{pmatrix}. $$

These matrices represent successive rotations of points on the sphere $\mathbb{S}^2$: first by $\vartheta_x$ around the x-axis, followed by $-\vartheta_y$ around the y-axis, and finally by $\vartheta_z$ around the z-axis. For a more comprehensive understanding of how matrices in $SO(d+1)$ operate on vectors in $\mathbb{S}^d$ for higher dimensions, see [98]. Additionally, the group $SO(d+1)$ acts transitively on $\mathbb{S}^d$, meaning that for any two points $\mathbf{x}_1, \mathbf{x}_2 \in \mathbb{S}^d$, there exists a matrix $\mathbf{u} \in SO(d+1)$ such that $\mathbf{u}\mathbf{x}_1 = \mathbf{x}_2$. This property highlights the versatility of $SO(d+1)$ in modeling rotational transformations on the sphere.
# Data availability

No data was used for the research described in the article.

# References

[1] W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Stat. 19 (1948) 293-325, http://dx.doi.org/10.1214/aoms/1177730196.

[2] P.R. Halmos, The theory of unbiased estimation, Ann. Math. Stat. 17 (1946) 34-43, http://dx.doi.org/10.1214/aoms/1177731020.

[3] A.J. Lee, U-Statistics: Textbooks and Monographs, vol. 110, Marcel Dekker, Inc., New York, 1990, theory and practice.

[4] H. Rubin, R.A. Vitale, Asymptotic distribution of symmetric statistics, Ann. Statist. 8 (1) (1980) 165-170, http://dx.doi.org/10.1214/aos/1176344898.

[5] R.J. Serfling, Approximation Theorems of Mathematical Statistics], in: Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1980, http://dx.doi.org/10.1002/9780470316481.

[6] E.B. Dynkin, A. Mandelbaum, Symmetric statistics, Poisson point processes and multiple Wiener integrals, Ann. Statist. 11 (3) (1983) 739-745, http://dx.doi.org/10.1214/aos/1176346241.

[7] J. Bretagnolle, Lois limites du bootstrap de certaines fonctionnelles, Ann. Inst. H. Poincaré Sect. B (N.S) 19 (3) (1983) 281-296.

[8] R. von Mises, On the asymptotic distribution of differentiable statistical functions, Ann. Math. Stat. 18 (1947) 309-348, http://dx.doi.org/10.1214/aoms/1177730385.

[9] M.A. Arcones, E. Giné, Limit theorems for U-processes, Ann. Probab. 21 (3) (1993) 1494-1542, http://dx.doi.org/10.1214/aop/1176989128.

[10] V.H. de la Peña, E. Giné, Decoupling, Springer-Verlag, New York, 1999, http://dx.doi.org/10.1007/978-1-4612-0537-1, from dependence to independence, Randomly stopped processes. U-statistics and processes. Martingales and beyond.

[11] S. Ghosal, A. Sen, A.W. van der Vaart, Testing monotonicity of regression, Ann. Statist. 28 (4) (2000) 1054-1082, http://dx.doi.org/10.1214/aos/1015956707.

[12] J. Abrevaya, W. Jiang, A nonparametric approach to measuring and testing curvature, J. Bus. Econom. Statist. 23 (1) (2005) 1-19, http://dx.doi.org/10.1198/07350010400000316.

[13] D. Nolan, D. Pollard, U-Processes: rates of convergence, Ann. Statist. 15 (2) (1987) 780-799, http://dx.doi.org/10.1214/aos/1176350374.

[14] R.P. Sherman, Maximal inequalities for degenerate U-processes with applications to optimization estimators, Ann. Statist. 22 (1) (1994) 439-459, http://dx.doi.org/10.1214/aos/1176325377.

[15] M.A. Arcones, Y. Wang, Some new tests for normality based on U-processes, Statist. Probab. Lett. 76 (1) (2006) 69-82, http://dx.doi.org/10.1016/j.spl.2005.07.003.

[16] A. Schick, Y. Wang, W. Wefelmeyer, Tests for normality based on density estimators of convolutions, Statist. Probab. Lett. 81 (2) (2011) 337-343, http://dx.doi.org/10.1016/j.spl.2010.10.022.

[17] E.W. Frees, Infinite order U-statistics, Scand. J. Stat. 16 (1) (1989) 29-45, http://dx.doi.org/10.1080/10485259508832638.

[18] C. Heilig, D. Nolan, Limit theorems for the infinite-degree U-process, Statist. Sinica 11 (1) (2001) 289-302.

[19] I. Soukarieh, S. Bouzebda, Renewal type bootstrap for increasing degree U-process of a Markov chain, J. Multivariate Anal. 195 (2023) http://dx.doi.org/10.1016/j.jmva.2022.105143, Paper No. 105143.

[20] W. Peng, T. Coleman, L. Mentch, Rates of convergence for random forests via generalized U-statistics, Electron. J. Stat. 16 (1) (2022) 232-292, http://dx.doi.org/10.1214/21-ejs1958.

[21] G.B. Cybis, M. Valk, S.R.C. Lopes, Clustering and classification problems in genetics through U-statistics, J. Stat. Comput. Simul. 88 (10) (2018) 1882-1902, http://dx.doi.org/10.1080/00949655.2017.1374387.

[22] F. Lim, V.M. Stojanovic, On U-statistics and compressed sensing I: Non-asymptotic average-case analysis, IEEE Trans. Signal Process. 61 (10) (2013) 2473-2485, http://dx.doi.org/10.1109/TSP.2013.2247598.

[23] S. Jadhav, S. Ma, An association test for functional data based on Kendall's tau, J. Multivariate Anal. 184 (2021) 9, http://dx.doi.org/10.1016/j.jmva.2021.104740, Paper (104740).

[24] D.Z. Bello, M. Valk, G.B. Cybis, Towards U-statistics clustering inference for multiple groups, J. Stat. Comput. Simul. 94 (1) (2024) 204-222, http://dx.doi.org/10.1080/00949655.2023.2239978.

[25] G.M. Szini, On the use of u-statistics for linear dyadic interaction models, 2023, arXiv:2309.02089.

[26] I. Kim, A. Ramdas, Dimension-agnostic inference using cross U-statistics, Bernoulli 30 (1) (2024) 683-711, http://dx.doi.org/10.3150/23-bej1613.

[27] S. Janson, Asymptotic normality for *m*-dependent and constrained *U*-statistics, with applications to pattern matching in random strings and permutations, Adv. in Appl. Probab. 55 (3) (2023) 841-894, http://dx.doi.org/10.1017/apr.2022.51.

[28] T. Le Minh, *U*-Statistics on bipartite exchangeable networks, ESAIM Probab. Stat. 27 (2023) 576-620, http://dx.doi.org/10.1051/ps/2023010.

[29] L. Chen, A.T.K. Wan, S. Zhang, Y. Zhou, Distributed algorithms for U-statistics-based empirical risk minimization, J. Mach. Learn. Res. 24 (2023) 43, Paper (263).

[30] K.K. Sudheesh, S. Anjana, M. Xie, U-statistics for left truncated and right censored data, Statistics 57 (4) (2023) 900-917, http://dx.doi.org/10.1080/02331888.2023.2217314.

[31] S. Janson, A functional limit theorem for random graphs with applications to subgraph count statistics, Random Structures Algorithms 1 (1) (1990) 15-37, http://dx.doi.org/10.1002/rsa.3240010103.

[32] W. Stute, Conditional U-statistics, Ann. Probab. 19 (2) (1991) 812-825, http://dx.doi.org/10.1214/aop/1176990452.

[33] E.A. Nadaraja, On a regression estimate, Teor. Verojatnost. I Primen. 9 (1964) 157-159, http://dx.doi.org/10.1137/1109020.

[34] G.S. Watson, Smooth regression analysis, Sankhyā Ser. A 26 (1964) 359-372, http://www.jstor.org/stable/25049340.

[35] S. Bouzebda, N. Taachouche, Rates of the strong uniform consistency for the Kernel-type regression function estimators with generalKernels on manifolds, Math. Methods Statist. 32 (1) (2023) 27-80, http://dx.doi.org/10.3103/s1066530723010027.

[36] S. Bouzebda, N. Taachouche, Rates of the strong uniform consistency with rates for conditional *U*-statistics estimators with general kernels on manifolds, Math. Methods Statist. 33 (2) (2024) 95-153, http://dx.doi.org/10.3103/s1066530724700066.

[37] S. Bouzebda, N. Taachouche, On the variable bandwidth kernel estimation of conditional *U*-statistics at optimal rates in sup-norm, Phys. A 625 (2023) 72, http://dx.doi.org/10.1016/j.physa.2023.129000, Paper (129000).

[38] W. Stute, Symmetized NN-conditional *U*-statistics, Research developments in probability and statistics, VSP, Utrecht, 1996, pp. 231-237.

[39] S. Bouzebda, A. Nezzal, Uniform consistency and uniform in number of neighbors consistency for nonparametric regression estimates and conditional *U*-statistics involving functional data, Jpn. J. Stat. Data Sci. 9 (2) (2024) 431-533, http://dx.doi.org/10.1007/s42081-022-00161-3.

[40] S. Bouzebda, A. Nezzal, Uniform in number of neighbors consistency and weak convergence of *k*NN, empirical conditional processes and *k*NN conditional *U*-processes involving functional mixing data, AIMS Math. 9 (2) (2024) 4427-4550, http://dx.doi.org/10.3934/math.2024218.

[41] S. Bouzebda, A. Nezzal, Asymptotic properties of conditional *U*-statistics using delta sequences, Comm. Statist. Theory Methods 53 (13) (2024) 4602-4657, http://dx.doi.org/10.1080/03610926.2023.2179887.
[42] T. Baranyi, L. Gyorı, A. Ludmány, H.E. Coffey, Comparison of sunspot area data bases, Mon. Not. R. Astron. Soc. 323 (1) (2001) 223-230, http://dx.doi.org/10.1046/j.1365-8711.2001.04195.x, arXiv:https://academic.oup.com/mnras/article-pdf/323/1/223/3222082/323-1-223.pdf.

[43] N.A.J. Feng Gao, Xiang-Yu. Huang, H. Wang, Assimilation of wind speed and direction observations: results from real observation experiments, Tellus A: Dyn. Meteorol. Ocean. 67 (1) (2015) 27132, http://dx.doi.org/10.3402/tellus.a.v67.27132.

[44] E. García-Portugués, I. Van Keilegom, R.M. Crujeiras, W. González-Manteiga, Testing parametric models in linear-directional regression, Scand. J. Stat. 43 (4) (2016) 1178-1191, http://dx.doi.org/10.1111/sjos.12236.

[45] J.L. Scealy, A.H. Welsh, Regression for compositional data by using distributions defined on the hypersphere, J. R. Stat. Soc. Ser. B Stat. Methodol. 73 (3) (2011) 351-375, http://dx.doi.org/10.1111/j.1467-9868.2010.00766.x.

[46] E. García-Portugués, R.M. Crujeiras, W. González-Manteiga, Kernel density estimation for directional-linear data, J. Multivariate Anal. 121 (2013) 152-175, http://dx.doi.org/10.1016/j.jmva.2013.06.009.

[47] P. Hall, G.S. Watson, J. Cabrera, Kernel density estimation with spherical data, Biometrika 74 (4) (1987) 751-762, http://dx.doi.org/10.1093/biomet/74.4.751.

[48] T. Chang, Spherical regression with errors in variables, Ann. Statist. 17 (1) (1989) 293-306, http://dx.doi.org/10.1214/aos/1176347017.

[49] L.-P. Rivest, Spherical regression for concentrated Fisher-von Mises distributions, Ann. Statist. 17 (1) (1989) 307-317, http://dx.doi.org/10.1214/aos/1176347018.

[50] J.A. Cuesta-Albertos, A. Cuevas, R. Fraiman, On projection-based tests for directional and compositional data, Stat. Comput. 19 (4) (2009) 367-380, http://dx.doi.org/10.1007/s11222-008-9098-3.

[51] G. Boente, D. Rodriguez, W. González Manteiga, Goodness-of-fit test for directional data, Scand. J. Stat. 41 (1) (2014) 259-275, http://dx.doi.org/10.1111/sjos.12020.

[52] A. Pewsey, E. García-Portugués, Recent advances in directional statistics, TEST 30 (1) (2021) 1-58, http://dx.doi.org/10.1007/s11749-021-00759-x.

[53] S. Efromovich, Density estimation for the case of supersmooth measurement error, J. Amer. Statist. Assoc. 92 (438) (1997) 526-535, http://dx.doi.org/10.2307/2965701.

[54] D.M. Healy Jr., H. Hendriks, P.T. Kim, Spherical deconvolution, J. Multivariate Anal. 67 (1) (1998) 1-22, http://dx.doi.org/10.1006/jmva.1998.1757.

[55] P.T. Kim, J.-Y. Koo, Optimal spherical deconvolution, J. Multivariate Anal. 80 (1) (2002) 21-42, http://dx.doi.org/10.1006/jmva.2000.1968.

[56] P.T. Kim, D.S.P. Richards, Deconvolution density estimation on compact Lie groups, in: Algebraic Methods in Statistics and Probability (Notre Dame, in 2000), in: Contemp. Math., Amer. Math. Soc., vol. 287, Providence, RI, 2001, pp. 155-171, http://dx.doi.org/10.1090/conm/287/04784.

[57] J.M. Jeon, B.U. Park, I. Van Keilegom, Nonparametric regression on Lie groups with measurement errors, Ann. Statist. 50 (5) (2022) 2973-3008, http://dx.doi.org/10.1214/22-aaos2218.

[58] G. Fan, X. Zhang, L. Zhu, Independence test via mutual information in the presence of measurement errors, Stat. Comput. 34 (6) (2024) 21, http://dx.doi.org/10.1007/s11222-024-10502-9, Paper (192).

[59] G.-L. Fan, H.-Y. Liang, Y. Shen, Penalized empirical likelihood for high-dimensional partially linear varying coefficient model with measurement errors, J. Multivariate Anal. 147 (2016) 183-201, http://dx.doi.org/10.1016/j.jmva.2016.01.009.

[60] D.B. Rubin, Inference and missing data, Biometrika 63 (3) (1976) 581-592, http://dx.doi.org/10.1093/biomet/63.3.581, with comments by R.J.A. Little and a reply by the author.

[61] R.J.A. Little, D.B. Rubin, Statistical analysis with missing data, third ed., in: Wiley Ser. Probab. Stat, John Wiley & Sons, Hoboken, NJ, 2019, http://dx.doi.org/10.1002/9781119482260.

[62] J. Josse, J.P. Reiter, Introduction to the special section on missing data, Statist. Sci. 33 (2) (2018) 139-141, http://dx.doi.org/10.1214/18-STS332IN.

[63] R.J.A. Little, D.B. Rubin, Statistical analysis with missing data, second ed., in: Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2002, http://dx.doi.org/10.1002/9781119013563.

[64] R.J.A. Little, D.B. Rubin, Statistical analysis with missing data, second ed., in: Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2002, http://dx.doi.org/10.1002/9781119013563.

[65] G. Claeskens, N.L. Hjort, Model selection and model averaging, in: Cambridge Series in Statistical and Probabilistic Mathematics, vol. 27, Cambridge University Press, Cambridge, 2008, http://dx.doi.org/10.1017/CBO9780511790485.

[66] S. Seaman, J. Galati, D. Jackson, J. Carlin, What is meant by missing at random? Statist. Sci. 28 (2) (2013) 257-268, http://dx.doi.org/10.1214/13-sts415.

[67] F. Mealli, D.B. Rubin, Clarifying missing at random and related definitions and implications when coupled with exchangeability, Biometrika 102 (4) (2015) 995-1000, http://dx.doi.org/10.1093/biomet/asv035.

[68] G. Lu, J.B. Copas, Missing at random, Likelihood Ignorability Model. Complet. Ann. Stat. 32 (2) (2004) 754-765, http://dx.doi.org/10.1214/0090536040000001066.

[69] D.M. Farewell, R.M. Daniel, S.R. Seaman, Missing at random: a stochastic process perspective, Biometrika 109 (1) (2022) 227-241, http://dx.doi.org/10.1093/biomet/asab002.

[70] J.M. Jeon, I. Van Keilegom, Density estimation and regression analysis on hyperspheres in the presence of measurement error, Scand. J. Stat. (2024) 1-44, http://dx.doi.org/10.1111/sjos.12684.

[71] W. Han K. Atkinson, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, in: Lecture Notes in Mathematics, vol. 2044, Springer, Heidelberg, 2012, http://dx.doi.org/10.1007/978-3-642-25983-8.

[72] A. Terras, Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane, second ed., Springer, New York, 2013, http://dx.doi.org/10.1007/978-1-4614-7972-7.

[73] G.S. Chirikjian, Applied and numerical harmonic analysis, in: Stochastic models, Information Theory, and Lie groups, vol. 2, Birkhäuser/Springer, New York, 2012, http://dx.doi.org/10.1007/978-0-8176-4944-9, analytic methods and modern applications.

[74] J.J. Sakurai, J. Napolitano, Modern Quantum Mechanics, Cambridge University Press, 2020, http://dx.doi.org/10.1017/9781108587280.

[75] S. Efromovich, Density estimation for the case of supersmooth measurement error, J. Amer. Statist. Assoc. 92 (438) (1997) 526-535, http://dx.doi.org/10.2307/2965701.

[76] P.T. Kim, J.-Y. Koo, H.J. Park, Sharp minimaxity and spherical deconvolution for super-smooth error distributions, J. Multivariate Anal. 90 (2) (2004) 384-392, http://dx.doi.org/10.1016/j.jmva.2003.08.004.

[77] C.A. León, J.-C. Massé, L.-P. Rivest, A statistical model for random rotations, J. Multivariate Anal. 97 (2) (2006) 412-430, http://dx.doi.org/10.1016/j.jmva.2005.03.009.

[78] T. Sei, H. Shibata, A. Takemura, K. Ohara, N. Takayama, Properties and applications of Fisher distribution on the rotation group, J. Multivariate Anal. 116 (2013) 440-455, http://dx.doi.org/10.1016/j.jmva.2013.01.010.

[79] S. B. Vardeman Y. Qiu, A wrapped trivariate normal distribution and Bayes inference for 3-D rotations, Statist. Sinica 24 (2) (2014) 897-917.

[80] S. Nadarajah, Y. Zhang, Wrapped: An r package for circular data, PLoS One 12 (12) (2017) 1-26, http://dx.doi.org/10.1371/journal.pone.0188512.

[81] R. Chakraborty, B.C. Vemuri, Statistics on the Stiefel manifold: theory and applications, Ann. Statist. 47 (1) (2019) 415-438, http://dx.doi.org/10.1214/18-AOS1692.

[82] D. Belomestny, A. Goldensluger, Density deconvolution under general assumptions on the distribution of measurement errors, Ann. Statist. 49 (2) (2021) 615-649, http://dx.doi.org/10.1214/20-aaos1969.

[83] A. Delaigle, J. Fan, R.J. Carroll, A design-adaptive local polynomial estimator for the errors-in-variables problem, J. Amer. Statist. Assoc. 104 (485) (2009) 348-359, http://dx.doi.org/10.1198/jasa.2009.0114.
[84] J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems, Ann. Statist. 19 (3) (1991) 1257-1272, http://dx.doi.org/10.1214/ aos/1176348248.

[85] L. Stefanski, R.J. Carroll, Deconvoluting kernel density estimators, Statistics 21 (2) (1990) 169-184, http://dx.doi.org/10.1080/02331889008802238.

[86] H. Hendriks, Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions, Ann. Statist. 18 (2) (1990) 832-849, http://dx.doi.org/10.1214/aos/1176347628.

[87] J. Fan, Y.K. Truong, Nonparametric regression with errors in variables, Ann. Statist. 21 (4) (1993) 1900-1925, http://dx.doi.org/10.1214/aos/1176349402.

[88] H. Kalf, On the expansion of a function in terms of spherical harmonics in arbitrary dimensions, Bull. Belg. Math. Soc. Simon Stevin 2 (4) (1995) 361-380, http://projecteuclid.org/euclid.bbms/1103408694.

[89] P.T. Kim, J.-Y. Koo, Optimal spherical deconvolution, J. Multivariate Anal. 80 (1) (2002) 21-42, http://dx.doi.org/10.1006/jmva.2000.1968.

[90] P.T. Kim, On the characteristic function of the matrix von Mises-Fisher distribution with application to SO(N)-deconvolution, in: High Dimensional Probability, II (Seattle, WA, 1999), in: Progr. Probab., vol. 47, Birkhäuser Boston, Boston, MA, 2000, pp. 477-492, http://dx.doi.org/10.1007/978-1-4612- 1358-1_31.

[91] P.T. Kim, D.S.P. Richards, Deconvolution density estimation on compact Lie groups, in: Algebraic Methods in Statistics and Probability (Notre Dame, in, 2000), in: Contemp. Math., Amer. Math. Soc., vol. 287, Providence, RI, 2001, pp. 155-171, http://dx.doi.org/10.1090/conn/287/04784.

[92] S. Bouzebda, N. Taachouche, Nonparametric conditional U-statistics on Lie groups with measurement errors, J. Complexity (2025) 1-59.

[93] J.M. Jeon, B.U. Park, I. Van Keilegom, Nonparametric regression on Lie groups with measurement errors, Ann. Statist. 50 (5) (2022) 2973-3008.

[94] Y. Katznelson, An Introduction to Harmonic Analysis, third ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004, http://dx.doi.org/10.1017/CBO9781139165372.

[95] M.G. Kendall, A new measure of rank correlation, Biometrika 30 (1/2) (1938) 81-93, http://dx.doi.org/10.2307/2332226.

[96] W. Stute, Universally consistent conditional U-statistics, Ann. Statist. 22 (1) (1994) 460-473, http://dx.doi.org/10.1214/aos/1176325378.

[97] F.G. Tricomi, A. Erdélyi, The asymptotic expansion of a ratio of gamma functions, Pacific J. Math. 1 (1951) 133-142, http://projecteuclid.org/euclid.pjm/ 1102613160.

[98] Y. Yayli, The factorization of elements of SO(n) in terms of the Euler's angles, Comm. Fac. Sci. Univ. Ank. Ser. A₁ Math. Stat. 41 (1-2) (1992) 67-71, http://dx.doi.org/10.1501/Commua1_0000000498, (1994).