# Multivariate spatial conditional U-quantiles: a Bahadur-Kiefer representation

Salim Bouzebda, Nourelhouda Taachouche

► To cite this version:

Salim Bouzebda, Nourelhouda Taachouche. Multivariate spatial conditional U-quantiles: a Bahadur-Kiefer representation. Results in Applied Mathematics, 2025, 26, pp.100593. 10.1016/j.rinam.2025.100593. hal-05091480

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# Multivariate spatial conditional *U*-quantiles: a Bahadur–Kiefer representation

Salim Bouzebda<sup>1 ID *</sup>, Nourelhouda Taachouche

Université de Technologie de Compiègne, Laboratoire de Mathématiques Appliquées de Compiègne, 57 Avenue de Landshut, Compiègne, 60203, France

## ARTICLE INFO

MSC:

60F05
60F15
60G42
62F35
62G05
62G20
62H05
62H12

Keywords:

Generalized conditional spatial quantiles
Conditional *U*-statistics
Multivariate analysis
Nonparametric methods
Dispersion estimation
Strong convergence
Empirical process
VC-classes

## ABSTRACT

Quantiles constitute a core concept in probability theory and theoretical statistics, providing an indispensable instrument in a wide array of applications. Although the univariate notion of quantiles is intuitively clear and mathematically well established, extending this concept to a multivariate framework poses significant theoretical and practical challenges. A well-established approach to extending univariate quantiles to the multivariate setting is the spatial (or geometric) framework, whose empirical counterparts exhibit notable robustness and admit an elegant Bahadur–Kiefer representation. Independently, another generalization of univariate quantiles leads to *U*-quantiles, which naturally encompass classical estimators such as the Hodge–Lehmann estimator for a central tendency. In this study, we bridge these perspectives by introducing multivariate conditional spatial *U*-quantiles and deriving their corresponding Bahadur–Kiefer representation. This representation enables us to establish fundamental theoretical properties, including weak convergence and a law of the iterated logarithm. These results are proved under some standard structural conditions on the Vapnik–Chervonenkis classes of functions and some mild conditions on the model. The uniform limit theorems discussed in this paper are key tools for further developments in data analysis involving empirical process techniques.

## 1. Introduction

A commonly examined characteristic in multivariate data analysis is the mean vector. Nonetheless, the arithmetic mean may prove inadequate in certain situations, particularly when data are substantially skewed or contaminated by outliers, thus raising questions about its effectiveness as a measure of central tendency. In such settings, *quantiles*—valued for their robustness against extreme observations—provide a durable, nonparametric substitute for the mean [1]. Contemporary biomedical research increasingly encounters multivariate failure-time outcomes, whether in clinical trials featuring multiple event types or in genetic studies involving correlated failure times within clusters (e.g., among litter-matched mice). These applications motivate the development of nonparametric procedures for quantile estimation in multivariate failure-time contexts. Despite vigorous efforts, a universally accepted definition of the multivariate median has yet to materialize, largely because higher-dimensional spaces lack a natural way to rank observations. Over the decades, numerous definitions for multivariate medians, and by extension, multivariate quantiles, have been proposed. For a wide-ranging survey of these diverse formulations, see [2]. Even after half a century of research, the generalization of this vital concept beyond the univariate setting remains problematic, primarily due to the absence

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: salim.bouzebda@utc.fr (S. Bouzebda), nourelhouda.taachouche@outlook.fr (N. Taachouche).</footnote>
of a canonical ordering in $\mathbb{R}^k$ with $k \ge 2$. In [3] and [4], the authors presented a comprehensive study of coordinate-wise medians and quantiles. However, such coordinate-specific approaches fail to capture the full joint structure of the multivariate distribution and do not exhibit crucial geometric properties—notably rotational invariance. A seminal contribution by [5] showed that, under absolute continuity, any finite set of sample quantiles from a $k$-variate distribution converges in distribution to a multivariate normal vector. Building on this foundational insight, [6] inaugurated the study of multivariate $U$-quantiles by embedding quantile indices into $U$-statistic kernels. Concurrently, [7] introduced the notion of spatial quantiles, thereby extending univariate quantile-based methodologies to higher-dimensional settings and enabling the nonparametric study of location, scale, skewness, and kurtosis. A variety of further approaches subsequently emerged to transfer classical univariate quantile techniques – historically a cornerstone of nonparametric inference – into the multivariate domain (see [8] for a detailed overview). Particularly noteworthy is the class of spatial quantiles [9,10], whose asymptotic behavior parallels the Bahadur and Bahadur-Kiefer representations familiar from the univariate literature [11–17] and has been rigorously established in [10,18]. These representations frequently quantify, potentially uniformly over a range of quantiles, the degree to which a linear approximation captures the discrepancy between an empirical quantile estimator and its underlying theoretical target.

The field of multivariate quantiles has since given rise to numerous methodologies. Multiple formulations of multivariate quantiles have been proposed [3,10,19–21]. Conditional versions that use quantiles to characterize conditional distributions first appeared in [22], later refined for the conditional median [23] and more general conditional quantiles [24,25]. Fundamental issues of invariance and equivariance—vital in multivariate settings—were addressed by [26], who introduced quantile-oriented functionals that preserve core symmetries. In another development, [27] proposed partial quantiles, which are defined relative to a partial order, remain order-equivalent under order-preserving transformations, and extend the notion of the efficient frontier while enhancing robustness to outliers. Advancements in this area continue apace: [28] developed a multivariate Q-Q plot based on spatial quantiles, whereas [29] defined set-valued quantiles through univariate distribution families under a convex cone. A further line of research by [30] harnessed copula-based methods for nonparametric multivariate quantile estimation, offering flexible modeling of dependencies. In the Bayesian domain, [31] constructed a model-free Gibbs posterior for multivariate quantiles without recourse to priors or marginalization, while [32] examined Bayesian inference for multivariate medians and quantiles through Dirichlet process priors.

Recent advances in optimal transport theory have also enriched our understanding of multivariate ranks and quantiles. Building on [33,34] undertook a thorough study of the properties of multivariate rank and quantile functions and extended those results to empirical settings. In a related innovation, [35] generalized spatial ($L_1$) quantiles to $L_p$-quantiles, introducing broader classes of loss functions for increased modeling flexibility. Directional quantiles, introduced by [36], use univariate conditional quantiles along rays extending from a chosen center, thus furnishing both global and directional probability interpretations without requiring convexity. More recently, [37] proposed an integrated regression framework to estimate unconditional quantiles, $M$-quantiles, and expectiles for multivariate outcomes by deploying a multidimensional Huber function that balances robustness with efficiency. In parallel, there has been significant progress in extending Bahadur representations from conventional quantiles to $U$-quantiles [38,39]. Under this perspective, quantiles derive from the ordered values of a real-valued kernel function $h(x_1, \dots, x_m)$ evaluated over all distinct $m$-tuples of observations. A notable example uses the arithmetic mean as the kernel: taking the median of those mean values yields a generalized Hodges-Lehmann location estimator. This estimator coincides with the standard median when $m = 1$ and reduces to the traditional Hodges-Lehmann estimator for $m = 2$ [40,41]. Overall, the theory of multivariate quantiles constitutes an active and rapidly advancing domain that weaves together fundamental statistical principles with innovative applications. By continuing to refine its theoretical foundations and extend its methodological scope, this line of research is poised to deepen our understanding of high-dimensional data structures and broaden the reach of quantile-based analyses across diverse fields.

## 1.1. *Contributions of the present work*

An extension of the preceding analysis to conditional quantile processes proves both highly valuable for practical applications and notably demanding from a technical standpoint. We introduce a novel framework for multivariate conditional $U$-quantiles that extends the geometric (or spatial) quantiles of [42]. In particular, we establish a Bahadur-type linear representation for the geometric conditional $U$-quantile estimator and provide an explicit convergence rate for its remainder term, thereby yielding theoretical insights and immediate statistical applications. A principal result of our study is the derivation of convergence rates for the modulus of continuity of the conditional empirical process. These asymptotic rates hinge on the complexity of the underlying function-index class, quantified via metric entropy. Our analysis culminates in proving both weak convergence and the law of the iterated logarithm for the proposed estimator. Moreover, this theoretical foundation – building on techniques akin to those in [42] – facilitates the construction of confidence ellipsoids for geometric conditional $U$-quantiles. In developing our main results, we employ sophisticated exponential inequalities for $U$-processes in concert with modern empirical process methodologies, enabling a refined understanding of the estimator's asymptotic properties. Notably, this paper opens several new research avenues, as described in the concluding section.

## 1.2. *Article structure*

This paper is organized as follows. In the next section, we introduce the fundamental concepts and mathematical tools necessary for presenting our main results. Section 2 establishes the essential theoretical background to ensure the paper remains as self-contained as possible. In Section 3, we describe the kernel-based estimation methods employed in our analysis. Our principal result,
which concerns the Bahadur-Kiefer representation for spatial conditional U-quantiles, is developed in Section 4. A key consequence of this result is the weak convergence of the estimator, which is discussed in Section 5. Furthermore, the law of the iterated logarithm (LIL) is addressed in Section 6. Section 7 provides a summary of the main findings and highlights their broader implications. To maintain the clarity of the exposition, detailed proofs and technical derivations are deferred to Section 8. Additionally, the Appendix includes further auxiliary results and technical lemmas that support our theoretical developments.

### 1.3. Notation

For a probability space $(T, \mathcal{T}, \mathbb{Q})$ and a measurable function $f: T \to \mathbb{R}$, we use the notation $\mathbb{Q}f := \int fd\mathbb{Q}$ whenever the integral is defined. For $q \in [1, \infty]$, let $\|\cdot\|_{\mathbb{Q},q}$ denote the $L^q(\mathbb{Q})$-seminorm, i.e., $\|f\|_{\mathbb{Q},q} := (\mathbb{Q}|f|^q)^{1/q} := (\int |f|^q d\mathbb{Q})^{1/q}$ for finite $q$ while $\|f\|_{\mathbb{Q},\infty}$ denotes the essential supremum of $|f|$ with respect to $\mathbb{Q}$. For a measurable space $(S, \mathcal{S})$ and a positive integer $r$, $S^r = S \times \dots \times S$ ($r$ times) denotes the product space equipped with the product $\sigma$-field $\mathcal{F}^r$. For a generic random variable $Y$ (not necessarily real-valued), let $\mathcal{L}(Y)$ denote the law (distribution) of $Y$. For $a, b \in \mathbb{R}$, let $a \vee b = \max\{a, b\}$ and $a \wedge b = \min\{a, b\}$. Let $\lfloor a \rfloor$ denote the integer part of $a \in \mathbb{R}$. "Constants" refer to finite, positive, and non-random numbers. We shall use also, throughout the paper, the notation

$$ \mathbf{X} = (\mathbf{X}_1, \dots, \mathbf{X}_m) \in \mathbb{R}^{pm} \quad \text{and} \quad \mathbf{X}_i := (\mathbf{X}_{i1}, \dots, \mathbf{X}_{i\ell}) \in \mathbb{R}^{d\ell}, \quad i \in I_n^m, $$

$$ \mathbf{Y} = (\mathbf{Y}_1, \dots, \mathbf{Y}_m) \in \mathbb{R}^{dm} \quad \text{and} \quad \mathbf{Y}_i := (\mathbf{Y}_{i1}, \dots, \mathbf{Y}_{i\ell}) \in \mathbb{R}^{d\ell}, \quad i \in I_n^m. $$

## 2. Preliminaries

### 2.1. Bahadur-Kiefer representation

Consider a sequence of independent and identically real valued variables $X_1, X_2, \dots$, all sharing a common distribution function $F(t) = P(X_1 \le t)$ for $t \in \mathbb{R}$. Suppose that $F(\cdot)$ is differentiable, with a bounded density function $f(\cdot) = F'(\cdot)$. For each $n \ge 1$, the empirical distribution function, based on the observations $X_1, \dots, X_n$, is given by

$$ \mathbb{F}_n(x) = n^{-1}\#\{X_i \le x : 1 \le i \le n\}, \quad \text{for } -\infty < x < \infty, $$

where # stands for cardinality. The corresponding empirical process $\{\mathbb{G}_n(x) : x \in \mathbb{R}\}$ based on the observations $X_1, \dots, X_n$ is then defined as:

$$ \mathbb{G}_n(x) := \sqrt{n} (\mathbb{F}_n(x) - \mathbb{F}(x)). \quad (1) $$

Next, the quantile function $\mathbb{Q}(\cdot)$ associated with $F(\cdot)$ is defined for $u \in (0, 1)$ by

$$ \mathbb{Q}(u) = \inf\{x : F(x) \ge u\}. $$

To extend this definition to the entire interval $[0, 1]$, we set

$$ \mathbb{Q}(0) = \lim_{t \downarrow 0} \mathbb{Q}(t) \quad \text{and} \quad \mathbb{Q}(1) = \lim_{t \uparrow 1} \mathbb{Q}(t). $$

The empirical quantile function, for each $n \ge 1$ and $u \in (0, 1)$, is defined analogously as

$$ \mathbb{Q}_n(u) = \inf\{x : \mathbb{F}_n(x) \ge u\}. $$

The empirical quantile process $\{q_n(u) : u \in [0, 1]\}$ based on the observations $X_1, \dots, X_n$ is defined by

$$ q_n(u) := \sqrt{n} f(\mathbb{Q}(u)) (\mathbb{Q}_n(u) - \mathbb{Q}(u)). $$

Let $U := U_1, U_2, \dots$ be i.i.d. rv's sitting on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. We let $U$ follow a uniform on $(0, 1)$ law, which is denoted by $U \stackrel{d}{=} \mathcal{U}(0, 1)$, where $\stackrel{d}{=}$ denotes the equality in distribution. Without loss of generality, we jointly define $X = X_1, X_2, \dots$ and $U = U_1, U_2, \dots$ on $(\Omega, \mathcal{A}, \mathbb{P})$, by setting $U = U_1 = \mathbb{F}(X_1), U_2 = \mathbb{F}(X_2), \dots$, or, equivalently, by continuity of $F(\cdot)$, $X = X_1 = \mathbb{Q}(U_1), X_2 = \mathbb{Q}(U_2), \dots$, for more details refer to [43]. For each $n \ge 1$, let $U_{n,1} \le \dots \le U_{n,n}$ be the order statistics of $U_1, \dots, U_n$, and set $U_{0,n} = 0$ and $U_{n+1,n} = 1$ for $n \ge 0$. Since the strict inequalities

$$ 0 = U_{0,n} < U_{1,n} < \dots < U_{n,n} < U_{n+1,n} = 1, \quad (2) $$

hold almost surely for all $n \ge 0$, and one can work, without loss of generality, on the event (2), of probability 1. Let

$$ \mathbb{E}_n(u) := n^{-1}\#\{U_i \le u : 1 \le i \le n\} = \mathbb{F}_n(\mathbb{Q}(u)), \quad \text{for } 0 \le u \le 1, $$

be the empirical df based upon $U_1, \dots, U_n$. Define the empirical qf, pertaining to $\mathbb{E}_n(\cdot)$, by

$$ \mathbf{V}_n(v) := \inf\{u \ge 0 : \mathbb{E}_n(u) \ge v\}, \quad \text{for } 0 \le v \le 1, $$

$\mathbf{V}_n(v) := 0, v < 0$, and $\mathbf{V}_n(v) := 1, v \ge 1$. The uniform empirical and quantile processes are defined respectively by $\alpha_n(x) = n^{1/2}(\mathbb{E}_n(x) - x)$ and $\beta_n(x) = n^{1/2}(\mathbf{V}_n(x) - x)$ for $x \in [0, 1]$, $\alpha_n(x) = \beta_n(x) = 0$ else. [44] first established an elegant asymptotic,
almost sure representation of a sample quantile for independent and identically distributed random variables with convergence rate $(O(n^{-3/4} (\log n)^{1/2} (\log \log n)^{1/4}))$, see also [45]. To be more precise, assuming that $(X_i)_{i=1}^n$ are i.i.d. and $f(Q(p)) > 0$, Bahadur first established the almost sure result

$$ \mathbb{Q}_n(p) = \mathbb{Q}(p) - \frac{p - \mathbb{F}_n(\mathbb{Q}(p))}{f(\mathbb{Q}(p))} + O_{\text{a.s.}} \left[ n^{-3/4} (\log n)^{1/2} (\log \log n)^{1/4} \right]. $$

[12,46] provided exact rates in the Bahadur representation for i.i.d. sequences. In particular, [12] showed that if $f'(\cdot)$ is bounded in a neighborhood of $\mathbb{Q}(p)$ and $f(\mathbb{Q}(p)) > 0$, then

$$ \limsup_{n \to \infty} \pm \frac{\mathbb{Q}_n(p) - \mathbb{Q}(p) - [p - \mathbb{F}_n(\mathbb{Q}(p))] / f(\mathbb{Q}(p))}{n^{-3/4}(\log \log n)^{3/4}} = \frac{2^{5/4} 3^{-3/4} p^{1/2} (1-p)^{1/2}}{f(\mathbb{Q}(p))} $$

almost surely for either choice of sign. The Bahadur-Kiefer representation (see [12,44] and e.g., [47]), by which, almost surely

$$ \limsup_{n \to \infty} n^{1/4} (\log n)^{-1/2} (\log \log n)^{-1/4} \| \alpha_n + \beta_n \| = 2^{-1/4}, $$

where $\|g\| = \|g\|_{\infty} = \sup_{0 \le x \le 1} |g(x)|$, refer also to [15,48]. [49] reviewed some of the most important large sample properties of empirical quantile processes and the associated Bahadur-Kiefer process. For recent references, we may refer to [50,51], and [52]. One significant extension of Bahadur's original result involves its application to U-quantiles [38,39]. These refer to the quantiles derived from the ordered evaluations of a real-valued kernel function, $h(x_1, \dots, x_m)$, applied to m-tuples of distinct indices selected from a sample $X_1, \dots, X_n$, where the total number of such evaluations is $n_{(m)} = n(n-1) \cdots (n-m+1)$. A fundamental example of this framework is given by the function:

$$ h(x_1, \dots, x_m) = \frac{x_1 + \dots + x_m}{m}, \qquad (3) $$

which, when combined with the median quantile, leads to a generalized form of the Hodges-Lehmann location estimator. Specifically, it represents the median of all possible m-wise averages of the sample observations for a chosen integer $m \ge 1$. Notable special cases include $m=1$, which corresponds to the conventional median, and $m=2$, which recovers the classical Hodges-Lehmann location estimator [40,41].

## 2.2. U-processes theory

First considered by [53] in connection with unbiased estimators and formally introduced by [54], U-statistics are fundamental tools in statistical theory. They are defined as follows: Given a sequence of random variables $\{W_i\}_{i=1}^{\infty}$ taking values in a measurable space $(S, \mathcal{S})$, and a measurable function $f: S^m \to \mathbb{R}$ (which is not necessarily symmetric), the U-statistic of order $m$ with kernel $f(\cdot)$ based on the sequence $\{W_i\}$ is defined by

$$ U_m^n(f, \mathbb{P}) := U_n(f) := \frac{(n-m)!}{n!} \sum_{(i_1, \dots, i_m) \in I_m^m} f(W_{i_1}, \dots, W_{i_m}), \quad \text{for } n \ge m, $$

where

$$ I_m^m = \{ (i_1, \dots, i_m) : i_j \in \mathbb{N}, 1 \le i_j \le n, i_j \ne i_k \text{ if } j \ne k \}, $$

and $\mathbb{N}$ denotes the set of positive integers. Here, $f(\cdot)$ is called the kernel of the U-statistic, operating on m-tuples of observations drawn from the sample without replacement. A function $f(\cdot)$ is said to be symmetric if, for any permutation $\pi$ of $\{1, 2, \dots, m\}$, the following holds:

$$ f(w_1, \dots, w_m) = f(w_{\pi(1)}, \dots, w_{\pi(m)}). $$

Common examples of estimators that can be expressed as U-statistics include the sample variance, Gini's mean difference, and Kendall's rank correlation coefficient. A classical test based on a U-statistic is the Wilcoxon signed-rank test for the hypothesis that the location parameter is zero (see, e.g., [55], Example 12.4). Several researchers, including [53,54,56], and [57], provided the first asymptotic results for U-statistics. In what follows $\mathcal{F}$ denotes a class of functions $f: S^m \to \mathbb{R}$ and $F$ is a measurable function with $F(x) \ge \sup_{f \in \mathcal{F}} |f(x)|$. We will need some measurability on the class $\mathcal{F}$. As usual, for any $\phi: \mathcal{F} \to \mathbb{R}$, we write $\|\phi(f)\|_{\mathcal{F}}$ for $\sup_{f \in \mathcal{F}} |\phi(f)|$. We study the behavior of the quantity

$$ \| U_m^n(f, \mathbb{P}) - \mathbb{P}^m f \|_{\mathcal{F}}, $$

where $\mathcal{F}$ is potentially an uncountable collection of symmetric functions $f: S^m \to \mathbb{R}$. As noted in [58, Page 138], the effectiveness of the randomization method critically depends on applying Fubini's theorem to expressions like

$$ \sup_{f \in \mathcal{F}} \left| \sum \varepsilon_{i_1} \cdots \varepsilon_{i_m} f(W_{i_1}, \dots, W_{i_m}) \right|. $$

Evaluating such integrals requires ensuring the measurability of these expressions, whether we integrate first over the random signs $\varepsilon$ and then over the variables $W$, or vice versa. When $\mathcal{F}$ is countable, measurability issues are straightforward and pose no problem. More generally, these issues can be circumvented – effectively treating $\mathcal{F}$ as if it were countable – when the sigma-algebra $\mathcal{S}$ is countably generated and contains singletons; that is, when $(S^m, \mathcal{S}^m)$ forms a separable measurable space. Moreover,
if $\mathcal{F}$ is an image-admissible Suslin class, there exists a mapping $\Psi(\cdot)$ from a Polish space $\mathcal{W}$ onto $\mathcal{F}$ such that the composition $(v, w_1, ..., w_m) \mapsto \Psi(v)(w_1, ..., w_m)$ is jointly measurable. The function classes of interest are often parameterized by subsets $\Theta \subset \mathbb{R}^d$, with the evaluation map being jointly measurable in both the variables and parameters. This underscores the importance of the image-admissible Suslin property in such contexts. As demonstrated in Theorem 10.3.2 of [59], if $\mathcal{F}$ is image-admissible Suslin, then for any constants $a$ and $b$, the function

$$ \left\| \sum_{(i_1, ..., i_m) \in I_n^m} a_{i_1, ..., i_m} f(w_{i_1}, ..., w_{i_m}) + b \mathbb{P}^m f \right\|_{\mathcal{F}} $$

is universally measurable with respect to any probability measure $\mathbb{P}$. Furthermore, for such a class $\mathcal{F}$, the map

$$ (v, w_1, ..., w_m) \mapsto \int \Psi(v)(w_1, ..., w_m) d\mathbb{P}^r(w_1, ..., w_r) $$

is jointly measurable for all $r \le m$, as established via the monotone class argument. The class $\mathcal{F}$ is assumed to be image admissible Suslin. For simplicity, we will refer to image admissible Suslin classes of functions over separable measurable spaces as measurable classes. Let $\mathbb{P}_n$ be the empirical measure associated with these random variables are defined as placing mass $1/n$ on each of the observations $W_i$, $i = 1, ..., n$, i.e.,

$$ \mathbb{P}_n := \frac{1}{n} \sum_{i=1}^{n} \delta_{W_i}. $$

Let $f: S \to \mathbb{R}$ be a measurable function. In the modern theory of the empirical, it is customary to identify $\mathbb{P}$ and $\mathbb{P}_n$ with the mappings given by

$$ f \to \mathbb{P}f = \int_S f d\mathbb{P}, \quad \text{and} \quad f \to \mathbb{P}_n f = \int_S f d\mathbb{P}_n = \frac{1}{n} \sum_{k=1}^{n} f(W_i). $$

Let us specify our setting better. For the sample $(\mathbf{X}_1, \mathbf{Y}_1), ..., (\mathbf{X}_n, \mathbf{Y}_n)$ having measure $\mathbb{P}$ and empirical measure $\mathbb{P}_n$, we define

$$ Z_n(f) := n^{1/2} (\mathbb{P}_n - \mathbb{P})(f), \quad f \in \mathcal{F}, $$

and put

$$ \sigma_0^2 = \sigma_0^2(\mathcal{F}) := \sup_{f \in \mathcal{F}} (\mathbb{P}(f^2) - \mathbb{P}(f)^2) := \sup_{f \in \mathcal{F}} \mathrm{Var}(f(\mathbf{X}_1, \mathbf{Y}_1)). $$

We introduce VC classes of functions as an adaptation of the Euclidean classes defined in [60], tailored to our setting. Let $\mathcal{F}$ be a family of measurable functions in $m$ variables, each symmetric in its arguments. For a pseudometric space $(T, d)$, the $\varepsilon$-covering number $N(T, d, \varepsilon)$ is given by

$$ N(T, d, \varepsilon) = \min \{ n : \text{there exists a cover of } T \text{ by } n \text{ balls of } d\text{-radius} \le \varepsilon \}. $$

Next, let $\mathcal{H}$ be a family of real-valued measurable functions on a measurable space $(V, \mathcal{W})$. Define

$$ \Xi_{\mathcal{H}}(v) := \sup \{|g(v)| : g \in \mathcal{H}\}. $$

A measurable function $Y(\cdot)$, finite almost everywhere on $(V, \mathcal{W})$, is called an envelope for $\mathcal{H}$ if $Y(v) \ge \Xi_{\mathcal{H}}(v)$ for every $v \in V$. Note that in some situations no such envelope may exist.

Assume $\mathcal{H}$ admits the envelope $Y(\cdot)$, and fix a positive real $\alpha$. We say that $\mathcal{H}$ is a Vapnik–Červonenkis class of functions for $Y(\cdot)$ and $\alpha$, denoted VC($Y, \alpha$), if there is a Lebesgue-integrable function $\lambda: [0, \infty) \to [0, \infty)$ satisfying

$$ [\log N(\mathcal{H}, \| \cdot \|_{L_2(\mu)} \tau(\mu Y^2)^{1/2})]^{\alpha/2} \le \lambda(\tau), \quad \tau \in \mathbb{R}_+, $$

for every probability measure $\mu$ with $\int Y^2 d\mu < \infty$. In fact, we will ultimately take $(V, \mathcal{W}) = (S^m, \mathfrak{S}^m)$ and $\alpha = m$. The family of VC classes is quite extensive. For example:

* VC classes of sets [59] fall under VC(1, $\alpha$) for any $\alpha > 0$.

* VC-subgraph (also referred to as VC-graph or polynomial) classes of functions [61] are VC($\Xi_{\mathcal{H}}^*, \alpha$) for all $\alpha$, provided $\Xi_{\mathcal{H}}$ admits a finite measurable envelope $\Xi_{\mathcal{H}}^*$.

* Whenever $\mathcal{H}$ is a Euclidean class for some envelope $Y$ in the sense of [60], it follows that $\mathcal{H}$ is VC($\Xi_{\mathcal{H}}$, $\alpha$) for every $\alpha > 0$.

# 3. Estimation procedures

## 3.1. Conditional quantiles

The concept of conditional quantile, which holds significant importance in statistics, has long been limited to probability measures over $\mathbb{R}$. Defining a suitable quantile concept in $\mathbb{R}^d$ presents a considerable challenge due to the lack of a canonical ordering in $\mathbb{R}^d, d > 1$, refer to [35]. One of the most successful multivariate quantile concepts is the concept of spatial (or geometric) quantiles
from [10]. More precisely, when $d \ge 2$, it is relatively straightforward to extend the geometric quantile to the geometric conditional quantile. For any vector $\mathbf{u} = (u_1, \dots, u_d)^\top$ in the open unit ball $B^{(d)} = \{\mathbf{u} | \mathbf{u} \in \mathbb{R}^d, \|\mathbf{u}\| < 1\}$ and any $\mathbf{l} \in \mathbb{R}^d$, denote by

$$ \Phi(\mathbf{u}, \mathbf{l}) = \|\mathbf{l}\| + \langle \mathbf{u}, \mathbf{l} \rangle, $$

where $\|\cdot\|$ is the usual Euclidean norm and $\langle \cdot, \cdot \rangle$ is the usual Euclidean inner product. Let $\mathbb{F}(y | \mathbf{x})$ denotes the conditional distribution of $\mathbf{Y}$ given $\mathbf{X} = \mathbf{x}$. Then, the $\mathbf{u}$-th geometric conditional quantile of $\mathbf{Y}$ given $\mathbf{X} = \mathbf{x}$ is defined as

$$ \mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x}) = \arg \min_{\theta \in \mathbb{R}^d} \mathbb{E} \{ \Phi(\mathbf{u}, \mathbf{Y} - \theta) - \Phi(\mathbf{u}, \mathbf{Y}) | \mathbf{X} = \mathbf{x} \}. $$

Equivalently, in terms of the spatial sign function (or unit vector function),

$$ S(\mathbf{u}) = \begin{cases} \frac{\mathbf{u}}{\|\mathbf{u}\|}, & \mathbf{u} \in \mathbb{R}^d, \mathbf{u} \neq \mathbf{0} \\ \mathbf{0}, & \mathbf{u} = \mathbf{0} \end{cases} $$

the quantile $\mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x})$ at $\mathbf{u}$ may be represented as the solution $\mathbf{x}$ of

$$ \mathbb{E}\{S(\mathbf{Y} - \mathbf{u}) | \mathbf{X} = \mathbf{x}\} = \mathbf{u}. \quad (4) $$

When $\|\mathbf{u}\|$ is close to one, we may think of $\mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x})$ as an “extreme” conditional quantile while a “central” conditional quantile corresponds to $\|\mathbf{u}\|$ being close to zero. Thus $\mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x})$, as indexed by a directional “outlyingness” parameter $\mathbf{u}$, measures quantitatively through $\|\mathbf{u}\|$ the extent of deviation from the center of the data cloud formed by $\mathbf{Y}$ given $\mathbf{X} = \mathbf{x}$; see, e.g., [7]. For a deeper understanding of the information carried by a geometric quantile, albeit defined differently than above, and for unconditional quantiles, we refer to [62,63, Section 2].

An estimate of the geometric conditional quantile $\mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x})$ is defined as follows through the empirical distribution. As usual, the nonparametric estimate of the conditional distribution $\mathbb{F}(y | \mathbf{x})$ is given by the so-called Nadaraya-Watson estimate as

$$ \mathbb{F}_n(\mathbf{y} | \mathbf{x}) = \sum_{i=1}^n w_{n,i} \mathbb{1}_{\{\mathbf{Y}_i \le \mathbf{y}\}}, $$

where the weight functions $w_{n,i}$ ($i = 1, 2, \dots, n$) are given by

$$ w_{n,i} = \frac{K_{h_n}(\mathbf{x} - \mathbf{X}_i)}{\sum_{i=1}^{n} K_{h_n}(\mathbf{x} - \mathbf{X}_i)}, \quad (5) $$

$K_{h_n}(\cdot) = (1/h_n^s) K(\frac{1}{h_n})$, $K(\cdot)$ is the kernel function, that is $K(\mathbf{x}) > 0$, $\sup_{\mathbf{x} \in \mathbb{R}^p} K(\mathbf{x}) = \kappa$ with $\kappa > 0$ and $\int K(t)dt = 1$ and $h_n > 0$ is the bandwidth tending at some rate that will be specified below. Let $\mathbb{1}_A$ denote the indicator function for set A. Then the estimator of the $\mathbf{u}$-th geometric conditional quantile $\mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x})$ is defined subsequently by

$$ \begin{aligned} \hat{\mathbb{Q}}_{\mathbb{F}_n}(\mathbf{u} | \mathbf{x}) &= \arg \min_{\theta \in \mathbb{R}^d} \int_{\mathbb{R}^d} \{\Phi(\mathbf{u}, \mathbf{y} - \theta) - \Phi(\mathbf{u}, \mathbf{y})\} \mathbb{F}_n(\mathrm{d}\mathbf{y} | \mathbf{x}) \\ &= \arg \min_{\theta \in \mathbb{R}^d} \sum_{i=1}^{n} w_{n,i} \{\Phi(\mathbf{u}, \mathbf{Y}_i - \theta) - \Phi(\mathbf{u}, \mathbf{Y}_i)\}. \end{aligned} $$

**Theorem 3.1 ([7]).** Under Conditions 1–7 in [7], $\hat{\mathbb{Q}}_{\mathbb{F}_n}(\mathbf{u} | \mathbf{x})$ has the following Bahadur-type representation

$$ \hat{\mathbb{Q}}_{\mathbb{F}_n}(\mathbf{u} | \mathbf{x}) - \mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x}) = D_{1n}^{-1} \sum_{i=1}^{n} w_{n,i} (S(\mathbf{Y} - \mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x})) + \mathbf{u}) + R_n, $$

with probability one, where

$$ D_{1n} = \mathbb{E} \left[ K_{h_n}(\mathbf{x} - \mathbf{X}) \mathfrak{F}(\mathbf{Y} - \mathbb{Q}_{\mathbb{F}}(\mathbf{u} | \mathbf{x})) \right] / \mathbb{E} K_{h_n}(\mathbf{x} - \mathbf{X}), $$

and

$$ \mathfrak{F}(\mathbf{y}) = (1/\|\mathbf{y}\|) (\mathfrak{J}_d - S(\mathbf{y})S^\top(\mathbf{y})) $$

for $\mathbf{y} \ne \mathbf{0}$, where $\mathfrak{F}(\mathbf{y})$ is the second order derivative of $\|\mathbf{y}\|$ and $\mathfrak{J}_d$ is the $d$-dimensional identity matrix. When $d \ge 3$, the remainder term $R_n = O((\log n) / (nh_n^s))$ and when $d = 2$, it is equal to $o((\log n) / (nh_n^s))^w$ for any $0 < w < 1$.

## 3.2. Conditional U-quantiles

For a sequence of random elements $\{(\mathbf{X}_i, \mathbf{Y}_i)\}_{i \in \mathbb{N}^*}$, where $\mathbf{X}_i \in \mathbb{R}^p$ and $\mathbf{Y}_i \in \mathcal{V}$, a Polish space, with $\mathbb{N}^* = \mathbb{N} \setminus \{0\}$. For a measurable vector-valued function $\varphi: \mathcal{V}^m \to \mathbb{R}^d$, let $\mathcal{Z} = \varphi(\mathbf{Y})$ given $\mathbf{X} = \tilde{\mathbf{t}}$ have conditional distribution $\mathbb{H}(\mathbf{z} | \tilde{\mathbf{t}})$ on $\mathbb{R}^d$ satisfying
**Assumption 1.**

(i) $\mathbb{H}(z | \tilde{t})$ has a density bounded on bounded subsets of $\mathbb{R}^d$ and

(ii) if $d \ge 2$, $\mathbb{H}(z | \tilde{t})$ is not concentrated on a line.

By substituting a random kernel evaluation for $X$ in (4), i.e., in terms of $Z \sim \mathbb{H}(\cdot | \tilde{t})$ and the sign function $S(\cdot)$ on $\mathbb{R}^d$, the conditional spatial U-quantile function corresponding to $\mathbb{H}(z | \tilde{t})$ is defined as the solution $z = \mathbb{Q}_{\mathbb{H}}(u | \tilde{t})$ of the equation

$$ \mathbb{E}\{S(z - Z) | X = \tilde{t}\} = u. \qquad (6) $$

As discussed in [10], for any conditional distribution function $\mathbb{H}(z | \tilde{t})$ the solution $\mathbb{Q}_{\mathbb{H}}(u | \tilde{t})$ to (6) always exists for any $u$ if condition (i) holds and is unique under the further condition (ii). The corresponding inverse function is thus

$$ \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(z | \tilde{t}) = \mathbb{E}\{S(z - Z) | X = \tilde{t}\}, $$

with sample analogue the vector-valued U-statistic

$$ \mathbb{Q}_{\phi, \lambda, \mathbb{H}_n}^{-1}(z | \tilde{t}) = \frac{\sum_{(i_1, \dots, i_m) \in I(m,n)} S(z - \varphi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_m})) K\left(\frac{\mathbf{t}_1 - \mathbf{X}_{i_1}}{\lambda h_n}\right) \dots K\left(\frac{\mathbf{t}_m - \mathbf{X}_{i_m}}{\lambda h_n}\right)}{\sum_{(i_1, \dots, i_m) \in I(m,n)} K\left(\frac{\mathbf{t}_1 - \mathbf{X}_{i_1}}{\lambda h_n}\right) \dots K\left(\frac{\mathbf{t}_m - \mathbf{X}_{i_m}}{\lambda h_n}\right)}, \qquad (7) $$

where

$$ I(m, n) = \{ (i_1, \dots, i_m) : 1 \le i_j \le n, i_j \ne i_r \text{ if } j \ne r \}, \qquad (8) $$

is the set of all $m$-tuples of distinct integers between 1 and $n$, $\lambda \in [\lambda_1, \lambda_2]$, $0 < \lambda_1 \le \lambda_2 < \infty$, and $\{h_n\}_{n \ge 1}$ is a sequence of positive constants converging to zero such that $nh_n^{pm} \to \infty$. Denote $K_h(t) = h^{-p}K(t/h)$, $t \in \mathbb{R}^p$ and let

$$ \tilde{K}(\tilde{t}) := \prod_{j=1}^{m} K(t_j), \quad \tilde{t} = (t_1, \dots, t_m) \in \mathbb{R}^{pm} $$

denote the product kernel. For a bandwidth $0 < h < 1$ and $\varphi \in \mathfrak{F}$, for some VC class of functions $\mathfrak{F}$, consider the U-kernel

$$ G_{\varphi, z, \lambda, \tilde{t}}(\cdot, \cdot) := (S(z - \varphi(\cdot))) \tilde{K}_{\lambda h}(\tilde{t} - \cdot), \quad z \in \mathbb{R}^d, \tilde{t} \in \mathbb{R}^{pm}, \lambda \in [\lambda_1, \lambda_2] \qquad (9) $$

and for the sample $(\mathbf{X}_1, \mathbf{Y}_1), \dots, (\mathbf{X}_n, \mathbf{Y}_n)$, define

$$ U_n(\varphi, z, \lambda, \tilde{t}) := U_m^n (G_{\varphi, z, \lambda, \tilde{t}}, \mathbb{P}) = \frac{(n-m)!}{n!} \sum_{i \in I_m^n} G_{\varphi, z, \lambda, \tilde{t}} (\mathbf{X}_i, \mathbf{Y}_i), $$

and

$$ f_n(\lambda, \tilde{t}) = \frac{(n-m)!}{n!} \sum_{i \in I_m^n} \tilde{K}_{\lambda h}(\tilde{t} - \mathbf{X}_i). $$

Let

$$ \mathcal{G} = \{ \lambda^m h^{pm} (S(z - \varphi(\cdot))) \tilde{K}_{\lambda h}(\tilde{t} - \cdot) : \lambda \in [\lambda_1, \lambda_2], \tilde{t} \in \mathbb{R}^{pm}, \varphi \in \mathfrak{F}, z \in \mathbb{R}^d \}. \qquad (10) $$

$$ \mathcal{G}^{(i)} = \{ \lambda^m h^{pm} (S(z - \varphi^{(i)}(\cdot))) \tilde{K}_{\lambda h}(\tilde{t} - \cdot) : \lambda \in [\lambda_1, \lambda_2], \tilde{t} \in \mathbb{R}^{pm}, \varphi \in \mathfrak{F}, z \in \mathbb{R}^d \}. \qquad (11) $$

Denote

$$ \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) = \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \cdot \prod_{j=1}^{m} f_{\mathbf{X}}(\mathbf{t}_j), \qquad (12) $$

and define the empirical processes

$$ \tilde{U}_n(\varphi, z, \lambda, \tilde{t}) = \sqrt{n\lambda h_n^p} (\mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}^{-1}(z | \tilde{t}) - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})), \qquad (13) $$

$$ \tilde{U}_n^*(\varphi, z, \lambda, \tilde{t}) = \sqrt{n\lambda h_n^p} (\mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}(z | \tilde{t}) - \mathbb{Q}_{\varphi, \mathbb{H}}(z | \tilde{t})). \qquad (14) $$

We can see that

$$ \tilde{U}_n(\varphi, z, \lambda, \tilde{t}) = \sqrt{n\lambda h_n^p} (\Psi_{n;1}(\varphi, z, \lambda, \tilde{t}) + \Psi_{n;2}(\varphi, z, \lambda, \tilde{t})) (1 + o_{\mathbb{P}}(1)), \qquad (15) $$

where

$$ \Psi_{n;1}(\varphi, z, \lambda, \tilde{t}) = \frac{1}{\mathbb{E}(f_n(\lambda, \tilde{t}))} (U_n(\varphi, z, \lambda, \tilde{t}) - \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})), $$
$$ \Psi_{n,2}(\varphi, z, \lambda, \tilde{t}) = - \frac{\mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})}{\mathbb{E}(f_n(\lambda, \tilde{t}))} \left( f_n(\lambda, \tilde{t}) - \prod_{j=1}^{m} f(t_j) \right). $$

To establish (15), we begin by recalling the representation

$$ \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}^{-1}(z | \tilde{t}) = \frac{U_n(\varphi, z, \lambda, \tilde{t})}{f_n(\lambda, \tilde{t})}, $$

which allows us to express

$$ \begin{aligned} \check{U}_n(\varphi, z, \lambda, \tilde{t}) &= \sqrt{n\lambda h_n^p} \left( \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}^{-1}(z | \tilde{t}) - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \right) \\ &= \sqrt{n\lambda h_n^p} \left( \frac{U_n(\varphi, z, \lambda, \tilde{t})}{f_n(\lambda, \tilde{t})} - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \right) \\ &= \sqrt{n\lambda h_n^p} \left[ \left( \frac{U_n(\varphi, z, \lambda, \tilde{t}) - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})}{f_n(\lambda, \tilde{t})} \right) + \left( \frac{\mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})}{f_n(\lambda, \tilde{t})} - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \right) \right]. \end{aligned} \quad (16) $$

For the first term in (16), by using the convergence $f_n(\lambda, \tilde{t}) \xrightarrow{\mathbb{P}} \prod_{j=1}^{m} f(t_j)$, and under suitable regularity conditions, we can approximate $\frac{1}{f_n(\lambda, \tilde{t})}$ by $\frac{1}{\mathbb{E}(f_n(\lambda, \tilde{t}))}$, we get that

$$ \frac{U_n(\varphi, z, \lambda, \tilde{t}) - \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})}{f_n(\lambda, \tilde{t})} \approx \frac{1}{\mathbb{E}(f_n(\lambda, \tilde{t}))} \left( U_n(\varphi, z, \lambda, \tilde{t}) - \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \right) + o_{\mathbb{P}}(1). $$

Denote by

$$ \Psi_{n;1}(\varphi, z, \lambda, \tilde{t}) := \frac{1}{\mathbb{E}[f_n(\lambda, \tilde{t})]} \left( U_n(\varphi, z, \lambda, \tilde{t}) - \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \right). \quad (17) $$

For the second term in (16), using (12), we rewrite

$$ \frac{\tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})}{f_n(\lambda, \tilde{t})} - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) = \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \cdot \left( \frac{\prod_{j=1}^{m} f(t_j)}{f_n(\lambda, \tilde{t})} - 1 \right) = - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t}) \cdot \frac{f_n(\lambda, \tilde{t}) - \prod_{j=1}^{m} f(t_j)}{f_n(\lambda, \tilde{t})}. $$

Again approximating $\frac{1}{f_n(\lambda, \tilde{t})}$ by $\frac{1}{\mathbb{E}(f_n(\lambda, \tilde{t}))}$, we define the second

$$ \Psi_{n;2}(\varphi, z, \lambda, \tilde{t}) := - \frac{\mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(z | \tilde{t})}{\mathbb{E}[f_n(\lambda, \tilde{t})]} \left( f_n(\lambda, \tilde{t}) - \prod_{j=1}^{m} f(t_j) \right). \quad (18) $$

By combining (17) and (18), we obtain the desired decomposition (15).

### 4. Bahadur-Kiefer representation for sample spatial conditional U-quantiles

Let us denote the weight generalizing (5) by

$$ W_{(i_1, \dots, i_m)}(\lambda, \tilde{t}) = \frac{K\left(\frac{\mathbf{t}_1 - \mathbf{X}_{i_1}}{\lambda h_n}\right) \cdots K\left(\frac{\mathbf{t}_m - \mathbf{X}_{i_m}}{\lambda h_n}\right)}{\sum_{(i_1, \dots, i_m) \in I(m,n)} K\left(\frac{\mathbf{t}_1 - \mathbf{X}_{i_1}}{\lambda h_n}\right) \cdots K\left(\frac{\mathbf{t}_m - \mathbf{X}_{i_m}}{\lambda h_n}\right)}. \quad (19) $$

Hence the corresponding extension of (12) in [42] to the conditional setting is given by

$$ \begin{aligned} \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}(\mathbf{u} | \tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}) &= -[D_1(\mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}))]^{-1} \sum_{(i_1, \dots, i_m) \in I(m,n)} [S(\mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}) - \varphi(\mathbb{Y})) - \mathbf{u}] \cdot W_{(i_1, \dots, i_m)}(\lambda, \tilde{\mathbf{t}}) + \mathbf{R}_n(\mathbf{u} | \tilde{\mathbf{t}}) \\ &= -[D_1(\mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}))]^{-1} \sum_{(i_1, \dots, i_m) \in I(m,n)} \left( [S(\mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}) - \varphi(\mathbb{Y}))] \cdot W_{(i_1, \dots, i_m)}(\lambda, \tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}})) \right) \\ &\quad + \mathbf{R}_n(\mathbf{u} | \tilde{\mathbf{t}}), \end{aligned} $$

where

$$ D_1(z, \tilde{t}) = \frac{\mathbb{E}[\tilde{K}_{\lambda h}(\tilde{t} - \mathbb{X})P(z - \mathbb{Z})]}{\mathbb{E}[\tilde{K}_{\lambda h}(\tilde{t} - \mathbb{X})]}, $$

where

$$ P(z - \mathbb{Z}) = \frac{\partial}{\partial z} \left( S(z - \mathbb{Z}) \right) = \frac{1}{\|z - \mathbb{Z}\|^2} \left[ I_d - \frac{1}{\|z - \mathbb{Z}\|^2} (z - \mathbb{Z})(z - \mathbb{Z})' \right]. $$
Note that $D_1(\mathbf{z}, \tilde{\mathbf{t}})$ is $d \times d$ symmetric and, unless $\mathbb{H}(\cdot | \tilde{\mathbf{t}})$ is supported by a straight line in $\mathbb{R}^d$, positive definite. For any compact $\mathfrak{B}_{\tilde{\mathbf{t}}} \subset \mathbb{W}_{\tilde{\mathbf{t}}} = \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbb{V} | \tilde{\mathbf{t}}) \subset \mathbb{B}^{d-1}$, let

$$ \Delta_{n,\lambda}(\mathfrak{B}_{\tilde{\mathbf{t}}} | \tilde{\mathbf{t}}) = \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{\mathbf{u} \in \mathfrak{B}_{\tilde{\mathbf{t}}}} \| \mathbf{R}_{n,\lambda}(\mathbf{u} | \tilde{\mathbf{t}}) \|. \quad (20) $$

### 4.1. Assumptions

In order to establish our results, we need the following hypotheses.

#### Assumption 2.

(i) For $\mathbf{r} \in \mathbb{N}$, (i) $\tilde{f}(\cdot) \in \mathcal{C}^r(\mathbb{R}^{pm})$ and (ii) $\mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbf{z} | \cdot) \in \mathcal{C}^r(\mathbb{R}^{pm})$.

(ii) $\tilde{f}(\cdot)$ is bounded away from zero on $\mathbb{R}^{pm}$.

(iii)

$$ \left| \int_{\mathbb{R}^{pm}} \sum_{\ell=0}^{r-1} \left( \begin{pmatrix} \mathbf{r} \\ \ell \end{pmatrix} \right) \sum_{l=1}^{r-1} \frac{1}{l!} (-1)^r \sum_{k_1+\cdots+k_m=\ell} \sum_{r_1+\cdots+r_m=r-\ell} \begin{pmatrix} \ell \\ k_1:m \end{pmatrix} \begin{pmatrix} \mathbf{r}-\ell \\ l_1:m \end{pmatrix} \times \prod_{j=1}^{m} \sum_{i_1, \ldots, i_{k_j}=1}^{m} \sum_{\tilde{t}_{1j}, \ldots, \tilde{t}_{ij}=1}^{m} \mathbf{s}_{i_1} \cdots \mathbf{s}_{i_{k_j}} \mathbf{s}_{\tilde{t}_{1j}} \cdots \mathbf{s}_{\tilde{t}_{ij}} \left| \frac{\partial^{k_j} f_{\mathbf{X}}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{k_j}}} (\mathbf{t}_j - \beta_{\mathbf{t},\mathbf{s}} \lambda h_n \mathbf{s}_j) \times \frac{\partial^{l_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{l_j}}} (\mathbf{z} | \tilde{\mathbf{t}} - \beta_{\mathbf{t},\mathbf{s}} \lambda h_n \tilde{\mathbf{s}}) \right| \prod_{j=1}^{m} |K(\mathbf{s}_j)| d\mathbf{s}_j \right| < \infty. \quad (21) $$

The main result to be proved here may now be stated precisely as follows.

**Theorem 4.1.** Let $\phi(\mathbf{y})$ be an $\mathbb{R}^d$-valued kernel with the property that, given $\mathbf{X} = \tilde{\mathbf{t}}$, its conditional cumulative distribution function $\mathbb{H}(\cdot | \tilde{\mathbf{t}})$ satisfies conditions (i) and (ii) in Assumption 1 and $nh_n^{p+2\kappa} \to 0$. Assume further that the inverse mapping $\mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbf{z} | \tilde{\mathbf{t}})$ is continuously differentiable, and that the function $D_1(\mathbf{z}, \tilde{\mathbf{t}})$ is locally Lipschitz continuous for all $\mathbf{z}$ in some open set $\mathbb{V} \subset \mathbb{R}^d$. Then, for any compact set $\mathfrak{B}_{\tilde{\mathbf{t}}} \subset \mathbb{W}_{\tilde{\mathbf{t}}} = \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbb{V} | \tilde{\mathbf{t}})$, there exist positive constants $\beta_1, \beta_2$, and $\beta_3$ such that the statistic $\Delta_{n,\lambda}(\mathfrak{B}_{\tilde{\mathbf{t}}} | \tilde{\mathbf{t}})$, as defined in (20), satisfies

$$ P^* \left\{ \Delta_{n,\lambda} (\mathfrak{B}_{\tilde{\mathbf{t}}} | \tilde{\mathbf{t}}) \ge \delta_n y \right\} \le \beta_1 \exp\left(-\frac{y}{\beta_2}\right), \quad (22) $$

for every $y > \beta_3 \log n$, where

$$ \delta_n = \begin{cases} (n\lambda_1 h_n^p)^{-3/4}, & \text{if } d=1, \\ (n\lambda_1 h_n^p)^{-1}(\log n)^{1/2}, & \text{if } d=2, \\ (n\lambda_1 h_n^p)^{-1}, & \text{if } d \ge 3. \end{cases} \quad (23) $$

The proof of Theorem 4.1 is postponed until Section 8. It is the most technically demanding, where we drew on ideas from the U-empirical process theory.

**Remark 4.2.** The rates of Bahadur-Kiefer type approximation vary across the cases $d > 2$, $d = 2$, and $d = 1$. Interestingly, the best approximation rates occur when $d > 2$, while the worst arise for $d = 1$. In the latter case, however, Bahadur-Kiefer theorems offer more precise insights. Nevertheless, we present our estimates – consistent with those obtained for $d > 1$ – for completeness.

Now, let $I = [c, d]$ be a compact interval in $\mathbb{R}$ with finite endpoints $(-\infty < c < d < \infty)$, and let $\eta > 0$ be a fixed positive constant. Define the extended interval by

$$ I^{\eta} = [c - \eta, d + \eta]. $$

As usual, we denote the marginal density function of $\mathbf{X}$ by $f_{\mathbf{X}}$. Next, consider the compact set

$$ J^m = I^\eta \times \dots \times I^\eta \subset \mathbb{R}^{pm}. $$

On this set, we define the class of functions

$$ \mathcal{Q}_{\phi, \mathbb{H}} = \{ \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbf{u} | \cdot) \tilde{f}(\cdot) : \phi \in \mathfrak{F}, \mathbf{u} \in \mathfrak{B}_{\tilde{\mathbf{t}}} \}, $$

where the function $\tilde{f}: \mathbb{R}^{pm} \to \mathbb{R}$ is given by

$$ \tilde{f}(\mathbf{t}) = \int f(\mathbf{t}_1, \mathbf{y}_1) \cdots f(\mathbf{t}_m, \mathbf{y}_m) d\mathbf{y}_1 \cdots d\mathbf{y}_m. $$

It follows directly that

$$ \tilde{f}(\tilde{\mathbf{t}}) = f_{\mathbf{X}}(\mathbf{t}_1) \cdots f_{\mathbf{X}}(\mathbf{t}_m). $$
**Theorem 4.3.** Suppose that the marginal density function $f_X$ of $X$ is continuous and strictly positive on the interval $J = I^n$, where $I$ is a compact interval and $\eta > 0$. Additionally, suppose that the class of functions $\mathcal{D}_H$ is uniformly equicontinuous and $h_n \to 0$ when $n \to \infty$. Under these conditions, it follows that

$$ \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{\mathbf{u} \in \mathcal{B}_i} \left| \mathbb{E} U_n(\varphi, \mathbf{u}, \lambda, \tilde{\mathbf{t}}) - \tilde{\mathbb{Q}}_{\varphi, f, H}^{-1}(\mathbf{u} | \tilde{\mathbf{t}}) \right| = o(1). \quad (24) $$

On the other hand, under condition (iii) of Assumption 2, we can derive that

$$ \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{\mathbf{u} \in \mathcal{B}_i} \left| \mathbb{E} U_n(\varphi, \mathbf{u}, \lambda, \tilde{\mathbf{t}}) - \tilde{\mathbb{Q}}_{\varphi, f, H}^{-1}(\mathbf{u} | \tilde{\mathbf{t}}) \right| = O(h_n^r). \quad (25) $$

The proof of Theorem 4.3 is postponed until Section 8.

## 5. Weak convergence results

As in [64], we say that the CLT (Central Limit Theorem) holds for the process $\{n^{1/2}(U_m^n(f, \mathbb{P}) - \mathbb{E}f) : f \in \mathcal{F}\}$ if there is a Gaussian process $\{G(f) : f \in \mathcal{F}\}$ which has a version with bounded and $d$-uniformly continuous paths, $d$ being the pseudodistance defined by $d^2(f, g) := \mathrm{Var}(\mathbb{P}^{m-1}(f - g))$, and if

$$ n^{1/2} (U_m^n(f, \mathbb{P}) - \mathbb{E}f) \xrightarrow{\mathcal{D}} \mathcal{G}(f) \text{ uniformly in } l^\infty(\mathcal{F}), \quad (26) $$

where convergence is in the sense of [65], (see e.g. [66]). Then $\{G(f) : f \in \mathcal{F}\}$ is a centered Gaussian process indexed by the class $\mathcal{F}$, with covariance

$$ \mathbb{E}G(f)G(g) = m^2 \mathbb{P} \left[ (\mathbb{P}^{m-1}f)(\mathbb{P}^{m-1}g) - m^2 (\mathbb{P}^m f)(\mathbb{P}^m g) \right]. $$

It is well-known that (26) is equivalent to both $(\mathcal{F}, d)$ being totally bounded and

$$ \lim_{\delta \to 0} \limsup_{n \to \infty} \mathbb{P} \left\{ \sup_{d(f,g) \le \delta} n^{1/2} \left| U_m^n(f, \mathbb{P}) - \mathbb{E}f - U_m^n(g, \mathbb{P}) + \mathbb{E}g \right| > \varepsilon \right\} = 0, \quad (27) $$

for all $\varepsilon > 0$ (where $\mathbb{P}$ is outer measure if the variable within the bracket is not measurable). If $\{G(f) : f \in \mathcal{F}\}$ has a version with bounded, $\rho$-uniformly continuous paths for some pseudodistance $\rho$ on $\mathcal{F}$, then (27) with $d$ replaced by $\rho$ is sufficient for the CLT (26). A key analytical technique employed herein is the Hoeffding decomposition, originally introduced in [54]. For completeness, we summarize its main components below.

### 5.1. Hoeffding decomposition

Consider a function $L$ of $m$ variables that is symmetric with respect to its arguments. For $1 \le k \le m$, the Hoeffding projections of $L$ with respect to a probability measure $\mathbb{P}$ are defined by

$$ \pi_k L(w_1, \dots, w_k) = (\delta_{w_1} - \mathbb{P}) \times \dots \times (\delta_{w_k} - \mathbb{P}) \times \mathbb{P}^{m-k}(L), $$

with the convention $\pi_0 L = \mathbb{E}[L(W_1, \dots, W_m)]$. Additional details can be found in [58]. The Hoeffding decomposition then asserts the following, which one can verify directly

$$ U_m^n(L, \mathbb{P}) - \mathbb{P}^m L = \sum_{k=1}^{m} \binom{m}{k} U_n^{(k)}(\pi_k L), \quad (28) $$

where, for a $k$-variate kernel $L$ with $1 \le k \le m$, we define

$$ U_k^n(L, \mathbb{P}) = \frac{(n-k)!}{n!} \sum_{i \in I_k^n} L(W_{i_1}, \dots, W_{i_k}). $$

Under the assumption that $L \in L_2(\mathbb{P}^m)$, the resulting decomposition is orthogonal, and

$$ \mathbb{E}[\pi_k L | W_2, \dots, W_k] = 0 \quad \text{for } k \ge 1. $$

This decomposition follows easily by expanding

$$ L(w_1, \dots, w_m) = \delta_{w_1} \times \dots \times \delta_{w_m} L = \left( \left( \delta_{x_1} - \mathbb{P} \right) + \mathbb{P} \right) \times \dots \times \left( \left( \delta_{x_m} - \mathbb{P} \right) + \mathbb{P} \right) L, $$

into terms of the form

$$ (\delta_{w_{i_1}} - \mathbb{P}) \times \dots \times (\delta_{w_{i_m}} - \mathbb{P}) \times \mathbb{P}^{m-k} L. $$

It is easy to verify that the symmetric function $L(\cdot)$ is $\mathbb{P}$-degenerate of order $r-1$, as defined below, if and only if $r = \min\{k > 0 : \pi_k L \neq 0\}$. In other words, the kernels $\pi_k L$ are canonical (completely degenerate or centered) with respect to $\mathbb{P}$. Moreover, the projections $\pi_k$, for $k \ge 1$, satisfy the nesting property

$$ \pi_k \circ \pi_l = \pi_k \quad \text{whenever } k \le l. $$
**Definition 5.1.** A $\mathbb{P}^m$-integrable symmetric function of $m$ variables, $L: S^m \to \mathbb{R}$, is $\mathbb{P}$-degenerate of order $r-1, 1 < r \le m$, if $\int L(w_1, \dots, w_m) d\mathbb{P}^{m-r+1}(w_r, \dots, w_m) = \int L d\mathbb{P}^m$ for all $w_1, \dots, w_{r-1} \in S$ whereas

$$ \int L(x_1, \dots, x_m) d\mathbb{P}^{m-r} (x_{r+1}, \dots, x_m) $$

is not a constant function. If $L$ is $\mathbb{P}^m$-centered and is $\mathbb{P}$-degenerate of order $m-1$, that is, if

$$ \int L(w_1, \dots, w_m) d\mathbb{P}(w_1) = 0 \quad \text{for all } w_2, \dots, w_m \in S, $$

then $L(\cdot)$ is said to be canonical or completely degenerate with respect to $\mathbb{P}$. If $L(\cdot)$ is not degenerate of any positive order, we say it is nondegenerate or degenerate of order zero.

Therefore, $L(\cdot)$ is degenerate of order $r-1 \ge 0$ iff its Hoeffding expansion, except for the constant term, starts at term $r$, that is,

$$ U_m^n(L, \mathbb{P}) - \mathbb{P}^m L = \sum_{k=r}^{m} \binom{m}{k} U_n^{(k)}(\pi_k L). $$

Considering $(X_1, Y_1), i \ge 1$, i.i.d.- $\mathbb{P}$ and assuming $L(\cdot)$ is in $L_2(\mathbb{P}^m)$, this is an orthogonal decomposition and

$$ \mathbb{E}[\pi_k L(X_k, Y_k) | (X_2, Y_2), \dots, (X_k, Y_k)] = 0, k \ge 1, $$

where we denote $X_k$ and $Y_k$ for $(X_1, \dots, X_k)$ and $(Y_1, \dots, Y_k)$, respectively. Thus, the kernels $\pi_k L$ are canonical for $\mathbb{P}$ (or completely degenerate, or completely centered). Also, $\pi_k, k \ge 1$, are nested projections, that is, $\pi_k \circ \pi_l = \pi_k$ if $k \le l$, and

$$ \mathbb{E}[(\pi_k L)^2 (X_k, Y_k)] \le \mathbb{E}[(L - \mathbb{E}L)^2(X, Y)] \le \mathbb{E}L^2(X, Y). \quad (29) $$

We introduce now a few more notations to state the asymptotic normality $\mathbb{Q}_{\mathbb{H}_n}^{-1}(z | \tilde{t})$. For $1 \le j, l, \ell \le m$ and $t_1, \dots, t_{3m} \in \mathbb{R}^{3pm}$, define

$$ \mathbb{Q}_{\phi, \mathbb{H}, j, l}^{-1}(z | \tilde{t}) $$

$$ := \mathbb{E} \left[ S(z - \varphi(Y_1, \dots, Y_{j-1}, Y, Y_{j+1}, \dots, Y_m)) S(z - \varphi(Y_{k+1}, \dots, Y_{k+l-1}, Y, Y_{m+l+1}, \dots, Y_{2m})) \right. \\ \left. \quad \middle| \quad X = t_j; X_i = t_i, \quad \forall i = 1, \dots, m, i \neq j; X_{m+i} = t_i, \quad \forall i = 1, \dots, m, i \neq l \right], $$

$$ \tilde{\mathbb{Q}}_{\phi, \mathbb{H}, j, l}^{-1}(z | t_1, \dots, t_{2m}) $$

$$ := \mathbb{E} \left[ S(z - \varphi(Y_1, \dots, Y_{j-1}, Y, Y_{j+1}, \dots, Y_m)) S(z - \varphi(Y_{k+1}, \dots, Y_{k+l-1}, Y, Y_{m+l+1}, \dots, Y_{2m})) \right. \\ \left. \quad \middle| \quad X = t_j; X_i = t_i, \quad \forall i = 1, \dots, 2m, i \notin \{j, m+l\} \right], $$

$$ \mathbb{Q}_{\phi, \mathbb{H}, j, l, \ell}^{-1}(z | t_1, \dots, t_{3m}) $$

$$ := \mathbb{E} \left[ S(z - \varphi(Y_1, \dots, Y_{j-1}, Y, Y_{j+1}, \dots, Y_m)) S(z - \varphi(Y_{k+1}, \dots, Y_{k+l-1}, Y, Y_{m+l+1}, \dots, Y_{2m})) \right. \\ \left. \quad S(z - \varphi(Y_{2m+1}, \dots, Y_{2m+\ell-1}, Y, Y_{2l+\ell+1}, \dots, Y_{3m})) \right. \\ \left. \quad \middle| \quad X = t_j; X_i = t_i, \quad \forall i = 1, \dots, 3m, i \notin \{j, k+l, 2m+\ell\} \right], $$

$$ \tilde{\mathbb{Q}}_{\phi, \mathbb{H}, j, l}^{-1}((z_1, z_2) | t_1, \dots, t_{2m}) $$

$$ := \mathbb{E} \left[ S(z_1 - \varphi(Y_1, \dots, Y_{j-1}, Y, Y_{j+1}, \dots, Y_m)) S(z_2 - \varphi(Y_{k+1}, \dots, Y_{k+l-1}, Y, Y_{m+l+1}, \dots, Y_{2m})) \right. \\ \left. \quad \middle| \quad X = t_j; X_i = t_i, \quad \forall i = 1, \dots, 2m, i \notin \{j, m+l\} \right]. $$

**Assumption 3.**

(i) $h_n \to 0$ and $n h_n^p \to \infty$;

(ii) $K(\cdot)$ is symmetric at 0, bounded, and compactly supported;

(iii) $\mathbb{Q}_{\phi, \mathbb{H}, j, l}^{-1}(z | \tilde{t})$ is continuous at $\tilde{t}$ for all $1 \le j, l \le k$.

(iv) $\mathbb{Q}_{\phi, \mathbb{H}}^{-1}(z | \tilde{t})$ is twice continuously differentiable in a neighborhood of $\tilde{t}$;

(v) $\mathbb{Q}_{\phi, \mathbb{H}, j, l, \ell}^{-1}(z | t_1, \dots, t_{3m})$ is bounded in a neighborhood of

$$ (t_1, \dots, t_m, t_{m+1}, \dots, t_{2m}, t_{2m+1}, \dots, t_m) \in \mathbb{R}^{3pm}, $$

for all $1 \le j, l, \ell \le m$;

(vi) $f_X(\cdot)$ is twice differentiable in neighborhoods of $t_i$ for all $1 \le i \le m$.

**Remark 5.2.** As in [67], our findings remain valid even when the assumption that the kernel function $K(\cdot)$ has compact support in Assumption 3 is replaced by the following alternative condition, denoted as (ii)’:
(ii)' There exists a sequence of positive real numbers {$a_n$} such that $a_n h_n^p$ converges to zero as $n$ approaches infinity, and

$$ \sqrt{n} \int_{\{\|v\| > a_n\}} |K(v)| dv \to 0 \quad \text{as } n \to \infty. $$

Below, we use the notation $V \stackrel{D}{=} \mathcal{N}(\mu, \sigma^2)$ to indicate that the random variable $V$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$.

**Proposition 5.3** (Asymptotic Normality of $Q_{\phi, \lambda, \mathbb{H}_n}^{-1}(z | \tilde{t})$; cf. Corollary 2.4 in [68]). Suppose **Assumption 3** is satisfied and $nh_n^{p+2} \to 0$ when $n \to \infty$. Then, for $i = 1, \dots, d$,

$$ \sqrt{n} h_n^p \left( Q_{\phi^{(i)}, \lambda, \mathbb{H}_n}^{-1}(z_i | \tilde{t}) - Q_{\phi^{(i)}, \mathbb{H}}^{-1}(z_i | \tilde{t}) \right) \stackrel{D}{\longrightarrow} \mathcal{N}(0, \rho_i^2), $$

where

$$ \rho_i^2 := \sum_{j,l=1}^k \mathbb{1}_{\{t_j=t_l\}} \left( Q_{\phi^{(i)}, \mathbb{H}, j, l}^{-1}(z_i | \tilde{t}) - Q_{\phi^{(i)}, \mathbb{H}}^{-1}(z_i | \tilde{t}) \right)^2 \frac{\|K\|_2^2}{f_X(t_j)}. $$

Moreover, let $\mathfrak{M}$ be any positive integer, and consider an array of points

$$ (t_1^{(1)}, \dots, t_m^{(1)}, \dots; t_1^{(\mathfrak{M})}, \dots, t_m^{(\mathfrak{M})}) \in \mathbb{R}^{pm \times \mathfrak{M}}. $$

Then, under analogous regularity conditions,

$$ \sqrt{n} h_n^p \left( Q_{\phi^{(i)}, \lambda, \mathbb{H}_n}^{-1} (z_{\ell}^{(i)} | t_1^{(i)}, \dots, t_m^{(i)}) - Q_{\phi^{(i)}, \mathbb{H}}^{-1} (z_{\ell}^{(i)} | t_1^{(i)}, \dots, t_m^{(i)}) \right)_{i=1, \dots, \mathfrak{M}} \stackrel{D}{\longrightarrow} \mathcal{N}(0, \mathfrak{S}), $$

where, for $1 \le \tilde{j}, \tilde{l} \le \mathfrak{M}$ and $1 \le \ell \le d$,

$$ [\mathfrak{S}]_{\tilde{j}, \tilde{l}, \ell} := \sum_{j,l=1}^k \mathbb{1}_{\{t_j^{(\tilde{j})}=t_l^{(\tilde{l})}\}} \left( \tilde{Q}_{\phi, \mathbb{H}, j, l}^{-1}((z_{\ell}^{(\tilde{j})}, z_{\ell}^{(\tilde{l})}) | t_1^{(\tilde{j})}, \dots, t_m^{(\tilde{j})}, t_1^{(\tilde{l})}, \dots, t_m^{(\tilde{l})}) \right. \\ \left. - Q_{\phi, \mathbb{H}}^{-1}(z_{\ell}^{(\tilde{j})} | t_1^{(\tilde{j})}, \dots, t_m^{(\tilde{j})}) Q_{\phi, \mathbb{H}}^{-1}(z_{\ell}^{(\tilde{l})} | t_1^{(\tilde{l})}, \dots, t_m^{(\tilde{l})}) \right) \frac{\|K\|_2^2}{f_X(t_j^{(\tilde{j})})}. \quad (30) $$

**Remark 5.4.** The second part of Proposition 5.3 follows from the first via an application of the Cramér–Wold theorem.

Define

$$ \mathfrak{W}_{\tilde{t}} = \mathbb{E} \left\{ \left[ S(\phi(\mathbb{Y}) - Q_{\phi, \mathbb{H}}(z | \tilde{t})) + z \right] \left[ S(\phi(\mathbb{Y}) - Q_{\phi, \mathbb{H}}(z | \tilde{t})) + z \right]^T \middle| X = \tilde{t} \right\}. $$

**Corollary 5.5.** Assume that **Assumption 1**, **2**, and **3** hold and $nh_n^{p+2} \to 0$ when $n \to \infty$. Then, as $n \to \infty$,

$$ \sqrt{n} h_n^p \mathfrak{W}_{\tilde{t}}^{-\frac{1}{2}} [D_1(Q_{\phi, \mathbb{H}}(z | \tilde{t}))] (Q_{\phi, \lambda, \mathbb{H}_n}^{-1}(z | \tilde{t}) - Q_{\phi, \mathbb{H}}^{-1}(z | \tilde{t})) \stackrel{D}{\longrightarrow} \mathcal{N}(0, \mathfrak{S}). $$

[64] considered some specific random distances for the weak convergence. Lemma 5.3.6 in [58], which is a straightforward application of the law of large numbers for $U$-processes, allow for the interchange of deterministic and random distances. The entropies arising from the random distance might not themselves be random variables. They were first introduced in the context of empirical processes by [69,70] and later by [71]. We recall the following distances:

$$ \tilde{e}_{n,p}(f,g) = \left( \sum_{i_1=1}^{J_n} \left| \sum_{j_n^{m}(i_1)} (f-g) (X_{i_1}, \dots, X_{i_m}) \right|^p \right)^{(1/p)\wedge 1}, $$

for $0 < p < \infty$. A direct application of Theorem 5.2.2. of [58] gives the following result.

**Proposition 5.6.** Let $\mathcal{G}$ be an image-admissible Suslin class of symmetric functions on $\mathbb{R}^d \times \mathbb{R}^{pm}$, and let $\mathbb{P}$ be a probability measure on the measurable space $(\mathbb{R}^d \times \mathbb{R}^p, \mathcal{B})$. Suppose further that $\mathcal{G}$ is of type $VC(\mathcal{G}, m)$. Then, as $n \to \infty$, the following convergence holds almost surely:

$$ \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{z \in \mathbb{R}^d, t \in \mathcal{I}^m} \left\| U_n(\phi^{(i)}, z_i, \lambda, \tilde{t}) - \mathbb{E}[U_n(\phi^{(i)}, z_i, \lambda, \tilde{t})] \right\| \to 0. $$

Furthermore, under condition (iii) of **Assumption 2** and $nh_n^{p+2\tau} \to 0$ when $n \to \infty$, we obtain

$$ \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{z \in \mathbb{R}^d, t \in \mathcal{I}^m} \left\| U_n(\phi^{(i)}, z_i, \lambda, \tilde{t}) - \tilde{Q}_{\phi^{(i)}, \mathbb{H}}^{-1}(z_i | \tilde{t}) \right\| \to 0 \quad \text{a.s.} $$

Invoking the above result along with **Theorem 4.3** immediately leads to the following corollary.
**Corollary 5.7.** Let $\mathcal{G}$ be an image-admissible Suslin class of symmetric functions on $\mathbb{R}^d \times \mathbb{R}^{p^m}$, and let $\mathbb{P}$ be a probability measure on $(\mathbb{R}^d \times \mathbb{R}^p, \mathcal{B})$. Under the assumptions of Proposition 5.6, one has

$$ \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{z \in \mathbb{R}^d, t \in \mathbb{I}^m} \left\| \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}^{-1}(\mathbf{z} | \tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbf{z} | \tilde{\mathbf{t}}) \right\| \to 0 \quad \text{a.s.} $$

By combining Corollary 5.7 with Theorem 4.1, we obtain the following statement.

**Corollary 5.8.** Let $\mathcal{G}$ be an image-admissible Suslin class of symmetric functions on $\mathbb{R}^d \times \mathbb{R}^{p^m}$, and let $\mathbb{P}$ be a probability measure on $(\mathbb{R}^d \times \mathbb{R}^p, \mathcal{B})$. Under the conditions of Proposition 5.6 and Assumption 1, the following almost-sure convergence holds:

$$ \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{z \in \mathbb{R}^d, t \in \mathbb{I}^m} \left\| \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}(\mathbf{z} | \tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{z} | \tilde{\mathbf{t}}) \right\| \to 0. $$

We next recall a result from Theorem 4.4 in [64]; see also Corollary 5.3.2 in [58].

**Proposition 5.9.** Assume that the image-admissible Suslin class of functions $\mathbb{P}^{m-1}\mathcal{G}^{(i)} := \{\mathbb{P}^{m-1}\varphi^{(i)} : \varphi^{(i)} \in \mathcal{G}^{(i)}\}$ is $\mathbb{P}$-Donsker and satisfies either

$$ \lim_{n \to \infty} \mathbb{E}^* \left[ \frac{1}{\sqrt{n}} \log N \left( \mathcal{G}^{(i)}, e_{n,2}, \delta/\sqrt{n} \right) \right]^2 = 0, \quad \forall \delta > 0, $$

or $\mathcal{G}$ is of type VC($\mathcal{G}$, $m$), then as $n \to \infty$,

$$ \sqrt{n} h_n^p \left( U_n(\varphi^{(i)}, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) - \mathbb{E} U_n(\varphi^{(i)}, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) \right) \to_{G_{\mathbb{P}} \circ \mathbb{P}^{m-1}} \text{in } \ell^{\infty}(\mathcal{G}^{(i)}), $$

uniformly for $\lambda \in [\lambda_1, \lambda_2]$.

By combining Proposition 5.9 with Theorem 4.3, the next corollary follows directly.

**Corollary 5.10.** Assume that the image-admissible Suslin class $\mathbb{P}^{m-1}\mathcal{G}^{(i)} := \{\mathbb{P}^{m-1}\varphi^{(i)} : \varphi^{(i)} \in \mathcal{G}^{(i)}\}$ is $\mathbb{P}$-Donsker and either

$$ \lim_{n \to \infty} \mathbb{E}^* \left[ \frac{1}{\sqrt{n}} \log N \left( \mathcal{G}^{(i)}, e_{n,2}, \delta/\sqrt{n} \right) \right]^2 = 0, \quad \forall \delta > 0, $$

or $\mathcal{G}$ is of type VC($\mathcal{G}$, $m$). If the condition (iii) of Assumption 2 holds and $nh_n^{p+2r} \to 0$ when $n \to \infty$, then, as $n \to \infty$,

$$ \sqrt{n} h_n^p \left( U_n(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}}) - \tilde{\mathbb{Q}}_{\varphi^{(i)}, \mathbb{H}}^{-1}(z_i | \tilde{\mathbf{t}}) \right) \to_{G_{\mathbb{P}} \circ \mathbb{P}^{m-1}} \text{in } \ell^{\infty}(\mathcal{G}^{(i)}), $$

uniformly for $\lambda \in [\lambda_1, \lambda_2]$.

**Corollary 5.11.** Assume that the image-admissible Suslin class $\mathbb{P}^{m-1}\mathcal{G}^{(i)} := \{\mathbb{P}^{m-1}\varphi^{(i)} : \varphi^{(i)} \in \mathcal{G}^{(i)}\}$ is $\mathbb{P}$-Donsker and either

$$ \lim_{n \to \infty} \mathbb{E}^* \left[ \frac{1}{\sqrt{n}} \log N \left( \mathcal{G}^{(i)}, e_{n,2}, \delta/\sqrt{n} \right) \right]^2 = 0, \quad \forall \delta > 0, $$

or $\mathcal{G}$ is of type VC($\mathcal{G}$, $m$). If condition (iii) of Assumption 2 is satisfied and $nh_n^{p+2r} \to 0$ when $n \to \infty$, then, for $\lambda \in [\lambda_1, \lambda_2]$,

$$ \check{U}_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) \to_{\tilde{G}_{\mathbb{P}} \circ \mathbb{P}^{m-1}} \text{in } \ell^{\infty}(\mathcal{G}^{(i)}), $$

where $\tilde{G}_{\mathbb{P}} \circ \mathbb{P}^{m-1}$ is a Gaussian process whose covariance is provided in (30).

Finally, we have yet another corollary under a similar Donsker-type requirement involving the $e_{n,1}$-metric.

**Corollary 5.12.** Assume that the image-admissible Suslin class $\mathbb{P}^{m-1}\mathcal{G}^{(i)} := \{\mathbb{P}^{m-1}\varphi^{(i)} : \varphi^{(i)} \in \mathcal{G}^{(i)}\}$ is $\mathbb{P}$-Donsker and either

$$ \lim_{n \to \infty} \mathbb{E}^* \left[ \frac{1}{\sqrt{n}} \log N \left( \mathcal{G}^{(i)}, e_{n,2}, \delta/\sqrt{n} \right) \right]^2 = 0, \quad \forall \delta > 0. $$

or $\mathcal{G}$ is of type VC($\mathcal{G}$, $m$). If Assumption 1 and condition (iii) of Assumption 2 hold and $nh_n^{p+2r} \to 0$ when $n \to \infty$, then uniformly for $\lambda \in [\lambda_1, \lambda_2]$,

$$ [D_1(\mathbb{Q}_{\varphi^{(i)}, \mathbb{H}}(\mathbf{z} | \tilde{\mathbf{t}}))] \check{U}_n^*(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}}) \to_{\tilde{G}_{\mathbb{P}} \circ \mathbb{P}^{m-1}} \text{in } \ell^{\infty}(\mathcal{G}^{(i)}). $$

# 6. The law of the iterated logarithm

In this section, we summarize the key definitions and notation primarily following [72,73], and [74]. Let $\delta_X - \mathbb{P}$ be a random variable, and define $H_{\mathbb{P}}$ as the reproducing kernel Hilbert space (RKHS) associated with $\delta_X - \mathbb{P}$. Concretely, $H_{\mathbb{P}}$ is the subspace of $\ell_{\infty}(\mathcal{G})$ consisting of functions of the form

$$ u_h(f) = \mathbb{E}[f(\mathbf{X}) h(\mathbf{X})], \quad f \in \mathcal{G}, $$
where $h$ belongs to the closed linear span in $L_2(\mathbb{P})$ of the collection

$$ \pi_{1,1}\mathcal{G} = \{f - \mathbb{P}f : f \in \mathcal{G}\}. $$

Equipping $\mathbb{P}$ with the inner product

$$ \langle u_{h_1}, u_{h_2} \rangle = \mathbb{P}[h_1 h_2], $$

the corresponding unit ball is given by

$$ K_{\mathcal{G}} = \{u_h \in H_{\mathbb{P}} : \|h\|_2 \le 1\}. $$

We say that a class $\mathcal{F}$ satisfies the law of iterated logarithm (LIL) if

$$ \left\{ \left( \frac{n}{2 \log \log n} \right)^{1/2} (\mathbb{P}_n - \mathbb{P}) f : f \in \mathcal{G} \right\} $$

is almost surely compact and if its limit set is $K_{\mathcal{G}}$. In the context of $U$-processes, we say that $\mathcal{G}$ satisfies the LIL provided that:

$$ \pi_1 \mathcal{G} \text{ satisfies the LIL,} $$

and

$$ \lim_{n \to \infty} \left\{ \frac{n}{\log \log n} \right\}^{1/2} \sup_{f \in \mathcal{G}} \left| U_m^n (\pi_k f, \mathbb{P}) \right| = 0, \quad \text{a.s. for } k = 2, \dots, m. $$

Under these conditions,

$$ \left\{ \left( \frac{n}{2 \log \log n} \right)^{1/2} \left( U_m^n(f, \mathbb{P}) - \mathbb{E}[f] \right) : f \in \mathcal{G} \right\} $$

is almost surely compact, and its limit set is $K_{\pi_1 \mathcal{G}}$.

Next, assuming the conditions of Proposition 5.9 and applying Theorem 2.2 of [74], we deduce that, for $i = 1, \dots, d$,

$$ \left\{ \left( \frac{n}{2 \log \log n} \right)^{1/2} \left( U_n(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}}) - \mathbb{E}[U_n(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}})] \right) : \varphi^{(i)} \in \mathcal{G}^{(i)} \right\} $$

is almost surely compact, and its limit set is $K_{\pi_1 \mathcal{G}^{(i)}}$. Furthermore,

$$ \sup_{\psi \in \mathcal{G}^{(i)}} \left\{ \frac{n}{\log \log n} \right\}^{1/2} \left| \left( U_n(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}}) - U_n(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}}) \right) - \frac{1}{n} \sum_{i=1}^{n} \pi_1 U_n(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}}) \right| = 0, \quad \text{a.s.} $$

Assuming also (iii) of Assumption 2 and $nh_n^{p+2r} \to 0$ when $n \to \infty$, we obtain

$$ \left\{ \left( \frac{n}{2 \log \log n} \right)^{1/2} \left( U_n(\varphi^{(i)}, z_i, \lambda, \tilde{\mathbf{t}}) - \tilde{\mathbb{Q}}_{\varphi^{(i)}, \mathbb{H}}^{-1}(z_i | \tilde{\mathbf{t}}) \right) : \varphi^{(i)} \in \mathcal{G}^{(i)} \right\} $$

is almost surely compact, and its limit set remains $K_{\pi_1 \mathcal{G}^{(i)}}$.

**Corollary.** Let $\mathbb{P}^{m-1}\mathcal{G}^{(i)} := \{\mathbb{P}^{m-1}\varphi^{(i)} : \varphi^{(i)} \in \mathcal{G}^{(i)}\}$ be an image-admissible Suslin class of functions that is $\mathbb{P}$-Donsker. If $\mathcal{G}$ is of type VC($\mathcal{G}$, $m$) and Assumption 1 along with (iii) of Assumption 2 hold and $nh_n^{p+2r} \to 0$ when $n \to \infty$, then

$$ \left\{ \left( \frac{n}{2 \log \log n} \right)^{1/2} \left[ \mathbf{D}_1(\mathbb{Q}_{\varphi^{(i)}, \mathbb{H}}(\mathbf{z} | \tilde{\mathbf{t}})) \right] \left( \mathbb{Q}_{\varphi^{(i)}, \lambda, \mathbb{H}_n}(z_i | \tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi^{(i)}, \mathbb{H}}(z_i | \tilde{\mathbf{t}}) \right) : \varphi^{(i)} \in \mathcal{G}^{(i)} \right\} $$

is almost surely compact, and its limit set is $K_{\pi_1 \mathcal{G}^{(i)}}$.

**Remark 6.1.** The proofs of our results can be readily adapted to demonstrate that they remain valid when the entropy condition is replaced by the bracketing condition. Specifically, there exist constants $C_0 > 0$ and $v_0 > 0$ such that

$$ \mathcal{N}_1(\mathcal{G}, L_2(\mathbb{P}), \epsilon) \le C_0 e^{-v_0}, \quad 0 < \epsilon < 1. $$

For the precise definition of the bracketing number $\mathcal{N}_1(\mathcal{G}, L_2(\mathbb{P}), \epsilon)$, the reader is referred to p. 270 of [55].

**Remark 6.2.** Let $\|\cdot\|$ denote a norm on $\mathbb{R}^p$, and define $B(x, r)$ as the closed ball centered at $x \in \mathbb{R}^p$ with radius $r$, i.e., the set of all points $z \in \mathbb{R}^p$ satisfying $\|x - z\| \le r$. For any integer $n \ge 1$ and $k \in \{1, \dots, n\}$, the $k$-nearest neighbor bandwidth at $x$, denoted by $\hat{k}_{n,x}$, is defined as the smallest radius $r \ge 0$ such that the ball $B(x, r)$ contains at least $k$ points from the sample $\{X_1, \dots, X_n\}$, i.e.,

$$ \hat{k}_{n,x} = \inf \left\{ r \ge 0 : \sum_{i=1}^{n} \mathbb{1}_{B(x,r)}(X_i) \ge k \right\}. $$

The $k$-nearest neighbor estimator is given by

$$ \mathbb{Q}_{\varphi, k, \mathbb{H}_n}^{-1}(\mathbf{z} | \tilde{\mathbf{t}}) = \frac{\sum_{(i_1, \dots, i_m) \in I(m,n)} S(\mathbf{z} - \varphi(\mathbf{Y}_{i_1}, \dots, \mathbf{Y}_{i_m})) \mathbb{1}_{B(t_1, \hat{k}_{n,t_1})} \cdots \mathbb{1}_{B(t_m, \hat{k}_{n,t_m})}}{\sum_{(i_1, \dots, i_m) \in I(m,n)} \mathbb{1}_{B(t_1, \hat{k}_{n,t_1})} \cdots \mathbb{1}_{B(t_m, \hat{k}_{n,t_m})}}. \quad (31) $$
This estimator represents an adaptive bandwidth variant of the Nadaraya-Watson estimator. The latter would be formulated similarly, except that it would employ a fixed bandwidth (typically of order $n^{-1/5}$ when $m = 1$) rather than the data-driven $\hat{k}_{n,x}$. A worthwhile direction for further investigation would be to analyze the behavior of $Q_{\phi,k,\mathbb{H}_n}^{-1}(z | \tilde{t})$ in the context of the k-nearest neighbor framework.

**Remark 6.3.** When the variables X and Y are independent, our setting simplifies to that considered in [42]. Consequently, up to a normalization factor, the limiting process coincides with the asymptotic behavior of the unconditional empirical process associated with the sequence Y, as expected. However, it is important to note that the normalization factor in this case is given by $(nh_n^p)^{1/2}$ with $h_n \to 0$, indicating a slower convergence rate. This reflects the additional cost incurred when making inferences about conditional (local) quantities.

**Remark 6.4.** In our framework, the choice of the kernel function is largely unrestricted, provided that it satisfies certain mild conditions, which will be specified subsequently. However, the selection of the bandwidth presents a greater challenge. It is important to emphasize that bandwidth selection plays a pivotal role in achieving a desirable rate of consistency, as it significantly impacts the bias of the estimator. Ideally, the chosen bandwidth should strike a balance between bias and variance to ensure optimal estimation performance. Consequently, a fixed bandwidth may not be suitable in all cases; instead, an adaptive bandwidth that varies according to specific criteria, the available data, and the location may be more appropriate. Classical methods often fail to accommodate such flexibility. For further insights and discussions on this topic, we refer the reader to [75–78]. Additionally, our results include uniform-in-bandwidth central limit theorems, which provide valuable practical guidance for selecting an optimal bandwidth.

**Remark 6.5.** As previously noted, a particular instance of the conditional U-statistic is the well-known Nadaraya-Watson estimator. Over the years, a substantial body of research has been dedicated to developing asymptotically optimal methods for selecting data-driven bandwidths for this estimator. Notable contributions in this domain include [77,79–82], and [83,84]. However, similar studies for the more general conditional U-statistic remain largely unexplored. Nonetheless, following the approach outlined in [85], we propose the following data-driven bandwidth selection method, which ensures consistency in estimation. This method extends a cross-validation procedure originally introduced by [80] for tuning the smoothing parameter of the Nadaraya-Watson estimator. For a given multi-index $i = (i_1, \dots, i_m) \in I_n^m$, define the index set:

$$I(m, n)(i) := \{k \in I(m, n) \mid k \neq i\} = I_n^m \setminus \{i\}.$$

Next, consider the estimator:

$$Q_{\phi, \lambda, \mathbb{H}_n, i}^{-1}(z | \mathbb{X}_i) = \frac{\sum_{(j_1, \dots, j_m) \in I(m,n)(i)} S(z - \varphi(Y_{j_1}, \dots, Y_{j_m})) K\left(\frac{\mathbf{X}_{i_1} - \mathbf{X}_{j_1}}{\lambda h_n}\right) \cdots K\left(\frac{\mathbf{X}_{i_m} - \mathbf{X}_{j_m}}{\lambda h_n}\right)}{\sum_{(j_1, \dots, j_m) \in I(m,n)(i)} K\left(\frac{\mathbf{X}_{i_1} - \mathbf{X}_{j_1}}{\lambda h_n}\right) \cdots K\left(\frac{\mathbf{X}_{i_m} - \mathbf{X}_{j_m}}{\lambda h_n}\right)} \quad (32)$$

The estimator in (32) is constructed by omitting the pair $(\mathbf{X}_i, \mathbf{Y}_i)$ and can be interpreted as a predictor for $S(z - \varphi(Y_{j_1}, \dots, Y_{j_m}))$ based on the observations $(\mathbf{X}_k, \mathbf{Y}_k)$ for $k \in I(m, n)(i)$. Let $w \ge 0$ be a measurable weighting function with support contained in $I$, and define the weighted squared difference between $S(z - \varphi(Y_{j_1}, \dots, Y_{j_m}))$ and its predictor $Q_{\phi, \lambda, \mathbb{H}_n, i}^{-1}(z | \mathbb{X}_i)$ as:

$$\mathrm{CV}(\lambda, \varphi, z) = \frac{(n-m)!}{n!} \sum_{i \in I_n^m} \left( S(z - \varphi(Y_{j_1}, \dots, Y_{j_m})) - Q_{\phi, \lambda, \mathbb{H}_n, i}^{-1}(z | \mathbb{X}_i) \right)^2 \hat{w}(\mathbb{X}_i),$$

where $\hat{w}(t) := \prod_{j=1}^m w(t_j)$. The optimal bandwidth is then selected as:

$$\hat{\lambda}_n := \lambda_n ((\mathbf{X}_1, \mathbf{Y}_1), \dots, (\mathbf{X}_n, \mathbf{Y}_n)) \in [\lambda_1, \lambda_2],$$

which minimizes, over $\lambda \in [\lambda_1, \lambda_2]$, the criterion:

$$\sup_{\varphi \in \mathcal{F}} \mathrm{CV}(\lambda, \varphi, z).$$

Since the selected bandwidth satisfies $\hat{\lambda}_n \in [\lambda_1, \lambda_2]$, Corollary 5.7 guarantees that:

$$\sup_{z \in \mathbb{R}^d, i \in I^m} \left\| Q_{\phi, \hat{\lambda}_n, \mathbb{H}_n}^{-1}(z | \tilde{t}) - Q_{\phi, \mathbb{H}_n}^{-1}(z | \tilde{t}) \right\| \to 0 \quad \text{a.s.}$$

**Remark 6.6.** It is well established that estimating a regression function (i.e., a conditional model) becomes particularly challenging when the dimension of the explanatory variable X is high. A notable consequence of this complexity is the slow convergence rate of the minimax-optimal estimator, given by $n^{-2k/(2k+d)}$, when estimating a regression function that is k times differentiable. Specifically, when the dimension d of X is large relative to k, this rate tends to zero at a slow pace. To address this well-known issue, often referred to as the curse of dimensionality, additional structural assumptions on the regression function must be
imposed. The simplest approach involves considering linear models, which provide a restrictive yet tractable parametric framework. However, this assumption can be relaxed in various ways. One alternative is to employ additive models, which simplify regression estimation by assuming that the function can be decomposed into a sum of lower-dimensional components. A further generalization is achieved through projection pursuit regression, where the regression function is modeled as a sum of univariate functions applied to projections of x onto different directions. A notable special case of this framework is the class of single-index models. For a more rigorous treatment of these techniques, we refer the reader to [86, Chapter 22]. Another promising direction involves semi-parametric models, which serve as an intermediate approach between linear and fully nonparametric models. These models aim to balance the flexibility of nonparametric methods with the interpretability and efficiency of parametric approaches. For an in-depth discussion on semi-parametric methods in the context of functional data, the interested reader may refer to the relevant literature.

**Remark 6.7.** The Value-at-Risk (VaR) metric has been widely accepted as a prevalent tool for evaluating market risk associated with either a single asset or a portfolio of assets. The Basel Committee on Banking Supervision has officially recognized this metric as a benchmark for assessing risk. Financial institutions utilize it for the objectives of managing assets and mitigating risks. Consider a sequence $\{X_i\}_{i=1}^n$ representing the market value of an asset across n time periods. Let $Y_i = \log(X_i/X_{i-1})$ denote the log-returns of the asset. Let us consider a sequence of random variables $\{Y_i\}_{i=1}^n$ that form a strictly stable dependent process. The individual random variables in this sequence have a marginal distribution function denoted by $F(\cdot)$. Let p be a positive value in close proximity to zero. The 1-p level Value at Risk (VaR) is defined as follows:

$$VaR_p = \inf\{x : F(x) \geq p\}.$$

This expression represents the minimum loss amount such that the chance of the market value loss exceeding VaR$_p$ is less than p. Hence, the examination of Value at Risk (VaR) is a distinct utilization of the p-quantile. For more elaboration, readers are encouraged to consult the following sources: [87,88], and the relevant references provided therein.

**Example 6.8 (Multivariate Generalized Hodges-Lehmann Scalar Spread Estimation, ([42], Example A)).** The work of [42] introduced a location estimator formulated as

$$HL_{(m)} = \text{spatial median} \left\{ \frac{\mathbf{X}_{l_1} + \dots + \mathbf{X}_{l_m}}{m} \right\},$$

which extends Eq. (3) to the multivariate setting and represents a U-quantile statistic, as elaborated in [89]. In this framework, denoting by $Q(\cdot)$ the spatial quantile function corresponding to the given kernel, it is possible to derive spread measures. An approach involves considering sample-based approximations of the volume of central regions defined as

$$\{Q(u) : \|u\| \le c\},$$

that generalize the concept of the interquartile range from univariate statistics to the multivariate case (see [89]; [21]; [90]). In our setting, a conditional version can be proposed

$$\{Q_{\phi,H}^{-1}(z | \tilde{t}) : \|z\| \le c\},$$

A comprehensive analysis of these measures will be undertaken in future research.

**Example 6.9 (Spatial U-Quantiles for Robust Dispersion Estimation, ([42], Example C)).** To estimate the conditional covariance matrix $\Sigma(\mathbf{t}_1, \mathbf{t}_2)$ associated with the conditional distribution $H(\cdot | (\mathbf{t}_1, \mathbf{t}_2))$, we consider the following conditional U-statistic estimator:

$$S_{\phi, \lambda, n}(\mathbf{t}_1, \mathbf{t}_2) = \frac{\sum_{1 \le i < j \le n} \phi(\mathbf{Y}_i, \mathbf{Y}_j) K_{\lambda h}(\mathbf{t}_1 - \mathbf{X}_i) K_{\lambda h}(\mathbf{t}_2 - \mathbf{X}_j)}{\sum_{1 \le i < j \le n} K_{\lambda h}(\mathbf{t}_1 - \mathbf{X}_i) K_{\lambda h}(\mathbf{t}_2 - \mathbf{X}_j)}, \quad (33)$$

where the matrix-valued kernel function is given by

$$\phi(\mathbf{y}_1, \mathbf{y}_2) = \frac{1}{2}(\mathbf{y}_1 - \mathbf{y}_2)(\mathbf{y}_1 - \mathbf{y}_2)^{\top}.$$

In a two-dimensional setting, this expression takes the explicit form:

$$\phi(\mathbf{y}_1, \mathbf{y}_2) = \frac{1}{2} \begin{pmatrix} (y_{11} - y_{21})^2 & (y_{11} - y_{21})(y_{12} - y_{22}) \\ (y_{11} - y_{21})(y_{12} - y_{22}) & (y_{12} - y_{22})^2 \end{pmatrix}.$$

However, this estimator is not robust to deviations from model assumptions or the presence of outliers. To overcome this limitation, we introduce an alternative measure of conditional dependence: the spatial median of the cumulative distribution function (cdf) $H(\cdot | (\mathbf{t}_1, \mathbf{t}_2))$, which corresponds to the spatial median of the random matrix $\phi(\mathbf{Y}_1, \mathbf{Y}_2)$ given that $(\mathbf{X}_1, \mathbf{X}_2) = (\mathbf{t}_1, \mathbf{t}_2)$. We denote this robust alternative by $\Sigma_{(2)}(\mathbf{t}_1, \mathbf{t}_2)$, and its empirical counterpart is given by

$$\hat{\Sigma}_{(2)}(\mathbf{t}_1, \mathbf{t}_2) = \text{spatial median} \{\phi(\mathbf{Y}_i, \mathbf{Y}_j) | (\mathbf{X}_1, \mathbf{X}_2) = (\mathbf{t}_1, \mathbf{t}_2)\}.$$

By construction, this estimator is robust to outliers and heavy-tailed distributions. Here, the spatial median of a random matrix is understood in terms of the spatial median of its vectorized form.
## 7. Concluding remarks and future works

A well-established method for extending univariate quantiles to multivariate settings involves the spatial (or geometric) perspective, renowned both for its robust sample analogs and its amenability to a straightforward Bahadur-Kiefer expansion. Meanwhile, univariate U-quantiles, such as those underlying the Hodges-Lehmann estimator for centrality, have similarly found success in more sophisticated contexts. In this paper, we unify these two research directions by proposing multivariate conditional spatial U-quantiles and deriving their associated Bahadur-Kiefer representations. This theoretical foundation paves the way for new nonparametric methods in multiple regression estimation.

Although tuning parameters for kernel-based estimators play a pivotal role in minimizing the mean squared error (MSE), we defer a detailed examination of such smoothing-parameter selection to future work. Established strategies like cross-validation or other optimal selection approaches are well-recognized in this regard; however, a comprehensive treatment of these methods falls outside our current scope.

In addition to smoothing-parameter considerations, controlling bias in kernel estimators remains a central challenge. Local linear fitting is known to offer potential improvements, but a deeper theoretical investigation is needed to illuminate the large-sample properties of such bias-correction procedures in a nonparametric setting. To the best of our knowledge, a rigorous analysis of bias-correction theory for the framework introduced here has yet to be undertaken. Addressing these issues promises to enhance the present methodology and bolster its performance when confronted with complex data structures.

A promising direction for extending this line of research involves examining how our approach may interact with other nonparametric estimators, including wavelet-based estimators [91,92], delta-sequence estimators [93], k-nearest neighbor estimators [94], and advanced local linear methods [95]. These techniques are vital in numerous applications and could benefit from the bias-correction ideas introduced here. Incorporating our proposed framework into these alternative estimators could foster productive synergies and further the development of nonparametric conditional quantile estimation.

Lastly, we suggest relaxing the assumption of stationarity (i.e., identical distributions) that underlies this study. Although stationarity is standard in many modeling scenarios, it can be inadequate for processes that evolve structurally over time; see [96]. Permitting locally stationary behaviors would enable the conditional quantile estimator to adapt dynamically, thereby accommodating nonstationary or time-varying environments. This extension naturally motivates the investigation of uniform limit theorems in nonstationary contexts, potentially revealing new insights into the asymptotic behavior of kernel estimators under more general conditions.

## 8. Mathematical development

In this section, we detail the principal results obtained in our investigation. To maintain coherence and facilitate understanding, the notation established earlier will be consistently employed throughout the exposition.

**Proof of Theorem 4.1.** Before proceeding with the proof of this theorem, we introduce some notation and present a fundamental lemma that is essential to our analysis.

Theorem 4.1 states that for any compact set $\mathfrak{B}_i \subset \mathbb{W}_i = \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbb{V} | \tilde{\mathbf{t}}) \subset \mathbb{B}^{d-1}$, the inequality given in (22) holds for

$$ \Delta_{n,\lambda}(\mathfrak{B}_i | \tilde{\mathbf{t}}) = \sup_{\lambda_1 < \lambda < \lambda_2} \sup_{\mathbf{u} \in \mathfrak{B}_i} \left\| \begin{aligned} & \mathbb{Q}_{\phi, \lambda, \mathbb{H}_n}(\mathbf{u} | \tilde{\mathbf{t}}) - \mathbb{Q}_{\phi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}) \\ & + \operatorname{inv}((\mathbb{Q}_{\phi, \mathbb{H}}^{-1})' \circ \mathbb{Q}_{\phi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}})) (\mathbb{Q}_{\phi, \lambda, \mathbb{H}_n}^{-1} - \mathbb{Q}_{\phi, \mathbb{H}}^{-1}) \circ \mathbb{Q}_{\phi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}) \end{aligned} \right\| \quad (34) $$

Let $v(g)$ denote $\int g dv$ for a signed measure $v$ and a measurable function $g$. We define the empirical spatial conditional U-quantile process as follows:

$$ \begin{aligned} \xi_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}, f) &:= \sqrt{n\lambda h_n^p} \frac{1}{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} \{U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) - \mathbb{E}U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}})\} \\ &:= \sqrt{n\lambda h_n^p} \{\tilde{U}_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}, f) - \mathbb{E}\tilde{U}_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}, f)\} \end{aligned} $$

Also, for open $\mathcal{W} \subset \mathcal{G} \times \mathbb{R}^d \times [\lambda_1, \lambda_2] \times \mathbb{R}^{pm}$ and function $\Psi(\cdot)$ from $\mathcal{W}$ into a normed linear space $(L, \| \cdot \|)$, we define

$$ \omega_{\Psi}(\rho; \delta) = \omega_{\Psi}^{\mathcal{W}}(\rho; \delta) := \sup \left\{ \| \Psi(s_1) - \Psi(s_2) \| : s_1, s_2 \in \mathbb{B}(\rho, \delta) \cap \mathcal{W} \right\}, \quad \rho \in \mathcal{W}, \delta > 0, $$

where $\mathbb{B}(\rho, \delta)$ denotes the open ball with center $\rho = (\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) \in \mathcal{G} \times \mathbb{R}^d \times [\lambda_1, \lambda_2] \times \mathbb{R}^{pm}$ and radius $\delta > 0$. Likewise, for compact $\mathfrak{B}_i \subset \mathcal{W}$, we put

$$ \omega_{\Psi}(\mathfrak{B}_i; \delta) = \omega_{\Psi}^{\mathcal{W}}(\mathfrak{B}_i; \delta) := \sup_{\rho \in \mathfrak{B}_i} \omega_{\Psi}(\rho; \delta), \delta > 0. $$

The following simple lemma will play an instrumental role in the sequel.
**Lemma 8.1.** For compact $\mathfrak{B} \subset W_{\tilde{\mathfrak{t}}} = \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbb{V} | \tilde{\mathfrak{t}}) \subset \mathbb{B}^{d-1}$, there exist $\delta_0 > 0$ and constants $d_1, d_2, d_3, d_4$ such that the condition

$$ (n\lambda_1 h_n^p)^{-1/2} \|\xi_n\|_{\mathbb{V}} + \left(\frac{n}{m}\right)^{-1} h_n^{-p} \le \delta_0, \qquad (35) $$

implies

$$ \Delta_{n,\lambda}(\mathfrak{B}_{\tilde{\mathfrak{t}}} | \tilde{\mathfrak{t}}) \le d_1(n\lambda_1 h_n^p)^{-1/2} \omega_{\xi_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{t}}}); d_2(n\lambda_1 h_n^p)^{-1/2} \|\xi_n\|_{\mathbb{V}} + d_3 \left(\frac{n}{m}\right)^{-1} h_n^{-p} \right) \\ + d_4(n\lambda_1 h_n^p)^{-1} \|\xi_n\|_{\mathbb{V}}^2 + \left(\frac{n}{m}\right)^{-1} h_n^{-p}, \quad n \ge m. $$

**Proof of Lemma 8.1.** By using the notation $\|g\|_C = \sup_{t \in C} |g(t)|$, we have

$$ \delta(\tilde{U}_n) := \|\tilde{U}_n - \mathbb{E}[\tilde{U}_n]\|_{\mathbb{V}} = (n\lambda_1 h_n^p)^{-1/2} \|\xi_n\|_{\mathbb{V}}. \qquad (36) $$

Since $\mathbb{H}(z | \tilde{\mathfrak{t}})$ is是非atomic, all sample evaluations $\phi(\mathbb{Y})$ are distinct and it readily follows that $\tilde{U}_n \circ \tilde{U}_n^{-1}(z)$ equals either $\tilde{U}_n(\phi(\mathbb{Y}))$, or $z$ according to whether $z$ does or does not belong to the closed ball of radius $\lambda_1 h_n^{-p} \binom{n}{m}^{-1}$ centered at $\tilde{U}_n(\phi(\mathbb{Y}))$. This leads to

$$ \|\tilde{U}_n \circ \tilde{U}_n^{-1}(z) - z\| \le \lambda_1 h_n^{-p} \binom{n}{m}^{-1}, $$

i.e., denoting the identity map from $\mathbb{V}$ to $\mathbb{V}$ by $\mathbf{I}_{\mathbb{V}}$, we have

$$ \eta(\tilde{U}_n, \tilde{U}_n^{-1}) := \|\tilde{U}_n \circ \tilde{U}_n^{-1} - \mathbf{I}_{\mathbb{V}}\|_{\mathbb{V}} \le \lambda_1 h_n^{-p} \binom{n}{m}^{-1}. \qquad (37) $$

Hence

$$ \|\tilde{U}_n - \mathbb{E}[\tilde{U}_n]\|_{\mathbb{V}} + \|\tilde{U}_n \circ \tilde{U}_n^{-1} - \mathbf{I}_{\mathbb{V}}\|_{\mathbb{V}} \le (n\lambda_1 h_n^p)^{-1/2} \|\xi_n\|_{\mathbb{V}} + \lambda_1 h_n^{-p} \binom{n}{m}^{-1}. $$

Now, by the assumption that $(\mathbb{Q}_{\phi, \mathbb{H}}^{-1})'$ is locally Lipschitz in $\mathbb{V}$, (36) yields that for compact $\mathfrak{B}_{\tilde{\mathfrak{t}}} \subset W_{\tilde{\mathfrak{t}}}$ and constant $\mathcal{C}$,

$$ \delta(\tilde{U}_n) \omega_{(\mathbb{Q}_{\phi, \mathbb{H}}^{-1})'}(\tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{t}}}); \mathcal{C}\delta(\tilde{U}_n)) \le \mathcal{C}\delta^2(\tilde{U}_n) \le \mathcal{C}(n\lambda_1 h_n^p)^{-1} \|\xi_n\|_{\mathbb{V}}^2, \qquad (38) $$

and also that $\tilde{U}_n$ is a diffeomorphism from $\mathbb{V}$ onto $W_{\tilde{\mathfrak{t}}} = \tilde{U}_n(\phi(\mathbb{Y}), \lambda, \tilde{\mathfrak{t}}, f)$. Consequently, a general inequality for functional inverse functions is given in [18] may be applied, with $\mathbf{G} = \tilde{U}_n$, $\mathbf{H} = \mathbb{E}[\tilde{U}_n]$ and $\mathbf{H}^{(-1)} = \tilde{U}_n^{-1}$, and yields via (38) the claim of the lemma. $\square$

For each $i = 1, \dots, d$, we consider the (real-valued) empirical process defined by

$$ W_n^{(i)}(f) := n^{1/2} (L_n^{(i)}(\cdot) - L^{(i)}(\cdot)) (f), \quad f \in \mathcal{G}^{(i)}, $$

over a suitable class $\mathcal{G}^{(i)}$ of real-valued $\mathbb{P}$-empirically measurable [97] functions on $\mathcal{Y}$, with $L_n^{(i)}$ the cdf of $(\mathbb{X}, \mathbb{Y})$ and $L_n^{(i)}$ its sample analogue. This is a $\mathbb{U}$-process indexed by $\{f \circ \phi^{(i)} : f \in \mathcal{G}^{(i)}\}$, as treated extensively in general in [64]. In particular, expressing the spatial sign function in $\mathbb{R}^d$ as $\mathbf{S} = (S^{(1)}, \dots, S^{(d)})$, we will use the classes $\mathcal{G}^{(i)}$, which is defined in (11), as well as the classes

$$ \mathcal{G}^{(i)} = \{f_1 - f_2 : f_1, f_2 \in \mathcal{G}^{(i)}\}, $$

i = 1, \dots, d. These are clearly uniformly bounded classes. Also, as pointed out in [18], they are Vapnik-Červonenkis subgraph classes ([98]; [99]), simply, VC-classes, let us say. For later reference, we note that if $\mathcal{G}$ is a VC-class, then so is $\mathcal{G} \circ \phi$ for a real-valued kernel $\phi(y_1, \dots, y_m)$ (see [98], Lemma 2.6.19). The probability inequalities we need for $W_n^{(i)}(\cdot)$ will draw upon ones of general interest that we develop separately in Appendix. These results, which will be applied in the proof of Theorem 1.1 below, will concern the metric $\rho_{\mathbb{P},2}$ defined on $\mathcal{G}$ by

$$ \begin{aligned} \rho_{\mathbb{P},2}(f,g) &:= (\mathbb{P}((f-g)^2) - (\mathbb{P}(f-g))^2)^{1/2} \\ &= (\mathrm{Var}(f-g) \circ \phi(\mathbb{Y}))^{1/2}, \quad f,g \in \mathcal{G}, \end{aligned} \qquad (39) $$

and the modulus defined for any function $Z : \mathcal{G} \to \mathbb{R}$ by

$$ \omega(Z; \rho_{\mathbb{P},2}; \delta) := \sup \left\{ |Z(f) - Z(g)| : f, g \in \mathcal{G}, \rho_{\mathbb{P},2}(f,g) \le \delta \right\}, \quad \delta > 0. $$

Our final preliminary is to state a result of [18] for metrics on $\mathbb{R}^d \times \mathbb{R}^{pm}$ of the form, for $\tilde{s}, \tilde{w}, \in \mathbb{R}^d$ and $\tilde{u}, \tilde{u}' \in \mathbb{R}^{pm}$,

$$ d_{\mathbb{P},2}(\tilde{s}, \tilde{w}, \tilde{u}, \tilde{u}') := \left( \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} \left\| S(\tilde{s} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - S(\tilde{w} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u}' - \tilde{v}}{\lambda h_n} \right) \right\|^2 d\mathbb{P}(\tilde{r}, \tilde{v}) \right)^{1/2}, \qquad (40) $$

for $\mathbb{P}$ a distribution on $\mathbb{R}^{pm} \times \mathbb{R}^d$.
**Lemma 8.2.** If $\mathbb{P}$ is a distribution on $\mathbb{R}^d$ with a bounded density, then there exists a constant $c_0 > 0$ such that for any $s \neq w \in \mathbb{R}^d$, $\tilde{\tau}_1, \tilde{\tau}_2 \in \mathbb{R}^{pm}$, and $\lambda' \in [\lambda_1, \lambda_2]$,

$$d_{\mathbb{P},2}(s, w, \tilde{\tau}_1, \tilde{\tau}_2) \le \Psi(\lambda', \lambda_1, p, m, \theta, \delta) + c_0 \begin{cases} \|s-w\|^{1/2} & \text{for } d=1, \\ \|s-w\| \log^{1/2} \frac{1}{\|s-w\|} & \text{for } d=2, \\ \|s-w\| & \text{for } d>2, \end{cases} \quad (41)$$

where

$$\Psi(\lambda', \lambda_1, p, m, \theta, \delta) = (\lambda' h_n)^{pm} \left( B_1\left(\frac{\delta}{\sqrt{pm}}\right) + B_2\left(\frac{\sigma}{\lambda_1} + 1\right) + 2\sqrt{B_1\left(\frac{\delta}{\sqrt{pm}}\right) B_2\left(\frac{\sigma}{\lambda_1} + 1\right)} \right), \quad (42)$$

with

$$B_1(\delta) = \sup_{\|\tilde{x}\| \le \delta} \int_{\mathbb{R}^{pm}} \left\| \tilde{K}(\tilde{u}) - \tilde{K}(\tilde{u} + \tilde{x}) \right\|^2 d\tilde{u},$$

$$B_2(\eta) = \sup_{1/\eta \le \rho \le \eta} \int_{\mathbb{R}^{pm}} \left\| \tilde{K}(\rho\tilde{u}) - \tilde{K}(\tilde{u}) \right\|^2 d\tilde{u},$$

which that

$$\lim_{\delta \to 0} B_1(\delta) = 0 \quad \text{and} \quad \lim_{\rho \to 1} B_2(\rho) = 0. \quad (43)$$

**Remark 8.3.** It is worth mentioning that when dealing with $U$-quantiles rather than conditional $U$-quantiles, the resulting distance corresponds to the one given in (41), but without the first term, as seen in [42].

**Proof of Lemma 8.2.** First, we remark that

$$\begin{align*}
& \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} \left\| S(\tilde{s} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - S(\tilde{w} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u}' - \tilde{v}}{\lambda' h_n} \right) \right\|^2 d\mathbb{P}(\tilde{r}, \tilde{v}) \\
&= \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} \left\| S(\tilde{s} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - S(\tilde{w} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) \right. \\
& \qquad \left. + S(\tilde{w} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - S(\tilde{w} - \tilde{r}) \tilde{K} \left( \frac{\tilde{u}' - \tilde{v}}{\lambda' h_n} \right) \right\|^2 d\mathbb{P}(\tilde{r}, \tilde{v}) \\
&= \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} \left\| (S(\tilde{s} - \tilde{r}) - S(\tilde{w} - \tilde{r})) \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) \right. \\
& \qquad \left. + S(\tilde{w} - \tilde{r}) \left( \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - \tilde{K} \left( \frac{\tilde{u}' - \tilde{v}}{\lambda' h_n} \right) \right) \right\|^2 d\mathbb{P}(\tilde{r}, \tilde{v}) \\
&\le \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} \left\| (S(\tilde{s} - \tilde{r}) - S(\tilde{w} - \tilde{r})) \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) \right\|^2 d\mathbb{P}(\tilde{r}, \tilde{v}) \\
&\quad + \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} \left\| S(\tilde{w} - \tilde{r}) \left( \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - \tilde{K} \left( \frac{\tilde{u}' - \tilde{v}}{\lambda' h_n} \right) \right) \right\|^2 d\mathbb{P}(\tilde{r}, \tilde{v}) \\
&\le \kappa^{2m} \int_{\mathbb{R}^d} \left\| S(\tilde{s} - \tilde{r}) - S(\tilde{w} - \tilde{r}) \right\|^2 d\mathbb{P}(\tilde{r}) \\
&\quad + \int_{\mathbb{R}^{pm}} \left\| \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - \tilde{K} \left( \frac{\tilde{u}' - \tilde{v}}{\lambda' h_n} \right) \right\|^2 d\tilde{v}. \tag{44}
\end{align*}$$

We get the same results for the first term in (44), as in [18]. For the second one, we denote, for $\lambda, \lambda' \in [\lambda_1, \lambda_2]$,

$$\Lambda_{\lambda, \lambda'} := \int_{\mathbb{R}^{pm}} \left\| \tilde{K} \left( \frac{\tilde{u} - \tilde{v}}{\lambda h_n} \right) - \tilde{K} \left( \frac{\tilde{u}' - \tilde{v}}{\lambda' h_n} \right) \right\|^2 d\tilde{v}.$$

To achieve the aimed result, choose $\lambda_1 = \gamma_1 < \dots < \gamma_L = \lambda_2$ such that, for $0 < \theta < 1$,

$$\max_{1 \le j \le L-1} (\gamma_{j+1} - \gamma_j) < \theta.$$

Therefore, by setting $\tilde{x} = \frac{\tilde{u}' - \tilde{v}}{\lambda' h_n}$ and applying the Cauchy–Schwarz inequality, we derive

$$\Lambda_{\lambda, \lambda'} = (\lambda' h_n)^{pm} \max_{1 \le j \le L-1} \int_{\mathbb{R}^{pm}} \left\| \tilde{K} \left( \frac{\tilde{u} - \tilde{u}'}{\gamma_j h_n} + \tilde{x} \frac{\gamma_{j+1}}{\gamma_j} \right) - \tilde{K}(\tilde{x}) \right\|^2 d\tilde{x}$$
$$
\begin{aligned}
&\le (\lambda' h_n)^{pm} \max_{1 \le j \le L-1} \left( \int_{\mathbb{R}^{pm}} \left\| \tilde{K} \left( \frac{\tilde{u} - \tilde{u}'}{\gamma_j h_n} + \tilde{x} \frac{\gamma_{j+1}}{\gamma_j} \right) - \tilde{K} \left( \frac{\tilde{x}}{\gamma_j} \frac{\gamma_{j+1}}{\gamma_j} \right) \right\|^2 d\tilde{x} \right. \\
&\qquad \left. + \int_{\mathbb{R}^{pm}} \left\| \tilde{K} \left( \frac{\tilde{x}}{\gamma_j} \frac{\gamma_{j+1}}{\gamma_j} \right) - \tilde{K} (\tilde{x}) \right\|^2 d\tilde{x} \right. \\
&\qquad \left. + 2 \int_{\mathbb{R}^{pm}} \left\| \tilde{K} \left( \frac{\tilde{u} - \tilde{u}'}{\gamma_j h_n} + \tilde{x} \frac{\gamma_{j+1}}{\gamma_j} \right) - \tilde{K} \left( \frac{\tilde{x}}{\gamma_j} \frac{\gamma_{j+1}}{\gamma_j} \right) \right\| \cdot \left\| \tilde{K} \left( \frac{\tilde{x}}{\gamma_j} \frac{\gamma_{j+1}}{\gamma_j} \right) - \tilde{K} (\tilde{x}) \right\| d\tilde{x} \right) \\
&\le B_1 \left( \frac{\delta}{\sqrt{pm}} \right) + B_2 \left( \frac{\theta}{\lambda_1} + 1 \right) + 2 \sqrt{B_1 \left( \frac{\delta}{\sqrt{pm}} \right) B_2 \left( \frac{\theta}{\lambda_1} + 1 \right)}.
\end{aligned}
$$

Thus, we conclude that

$$
\Lambda_{\lambda, \lambda'} \le (\lambda' h_n)^{pm} \left( B_1 \left( \frac{\delta}{\sqrt{pm}} \right) + B_2 \left( \frac{\sigma}{\lambda_1} + 1 \right) + 2 \sqrt{B_1 \left( \frac{\delta}{\sqrt{pm}} \right) B_2 \left( \frac{\sigma}{\lambda_1} + 1 \right)} \right).
$$

This completes the proof. $\square$

Continuation of the proof of Theorem 4.1

We thoughtfully align our approach with the methodology presented by [[18], Theorem 2.1] and [42], carefully adapting it to our framework. This involves substituting results for classical empirical processes with those tailored to conditional *U*-processes. By applying Theorem A.3, in Appendix, and inequality (65) componentwise to the vector-valued process $\tilde{\xi}_n$, we obtain that

$$
P^* (\|\tilde{\xi}_n\|_V \ge t) \le d_1 \exp \left( -\frac{t^2}{d_2^2} \right), \quad t > 0. \tag{45}
$$

From (39) and (40), we have, for $1 \le i \le d$,

$$
\begin{aligned}
& \rho_{L,2} \left( S^{(i)}(\mathbf{z}_1 - \cdot) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_1}{\lambda h_n} \cdot \right), S^{(i)}(\mathbf{z}_2 - \cdot) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_2}{\lambda h_n} \cdot \right) \right) \\
& = \left( \operatorname{Var} \left( S^{(i)}(\mathbf{z}_1 - \varphi) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_1}{\lambda h_n} \cdot \right) - S^{(i)}(\mathbf{z}_2 - \varphi) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_2}{\lambda h_n} \cdot \right) \right) \right)^{1/2} \\
& \le \left( \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} \left| S^{(i)}(\mathbf{z}_1 - \varphi) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_1}{\lambda h_n} \cdot \right) - S^{(i)}(\mathbf{z}_2 - \varphi) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_2}{\lambda h_n} \cdot \right) \right|^2 d\mathbb{L} \right)^{1/2} \\
& = d_{L,2}(\mathbf{z}_1, \mathbf{z}_2, \tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2), \quad \mathbf{z}_1, \mathbf{z}_2 \in \mathbb{R}^d \text{ and } \tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2 \in \mathbb{R}^{pm}.
\end{aligned}
$$

Put

$$
\bar{\delta} = \sup \{ d_{L,2}(\mathbf{z}_1, \mathbf{z}_2, \tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2) : \|(\mathbf{z}_1, \tilde{\mathbf{t}}_1) - (\mathbf{z}_2, \tilde{\mathbf{t}}_2)\| \le \bar{\delta} \}.
$$

Then, for $1 \le i \le d$, we get that

$$
\begin{aligned}
& \omega_{\tilde{\xi}_n}^{(i)} (\tilde{U}_n^{-1}(\mathfrak{B}\tilde{\mathbf{t}}); \delta) \\
& = \sup_{\tilde{w} \in \tilde{U}_n^{-1}(\mathfrak{B}\tilde{\mathbf{t}})} \left\{ \left| W_n^{(i)} (S^{(i)}(\mathbf{z}_1 - \cdot) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_1}{\lambda h_n} \cdot \right)) - W_n^{(i)} (S^{(i)}(\mathbf{z}_1 - \cdot) \tilde{K} \left( \frac{\tilde{\mathbf{t}}_2}{\lambda h_n} \cdot \right)) \right| \right. \\
& \qquad \left. : \mathbf{z}_1, \mathbf{z}_2 \in \mathbb{B}(\tilde{w}, \delta), \tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2 \in \mathbb{R}^{pm} \right\} \\
& \le \sup_{\tilde{w} \in \tilde{U}_n^{-1}(\mathfrak{B}\tilde{\mathbf{t}})} \left\{ \left| W_n^{(i)}(f_1) - W_n^{(i)}(f_2) \right| : f_1, f_2 \in \mathcal{G}^{(i)}, \rho_{L,2}(f_1, f_2) \right. \\
& \qquad \left. \le \sup_{\| (\mathbf{z}_1, \tilde{\mathbf{t}}_1) - (\mathbf{z}_2, \tilde{\mathbf{t}}_2) \| \le \delta} d_{L,2}(\mathbf{z}_1, \mathbf{z}_2, \tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2), \tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2 \in \mathbb{R}^{pm} \right\} \\
& =: \omega_n (W_n^{(i)}; \rho_{L,2}; \bar{\delta}).
\end{aligned}
\tag{46}
$$

Using Lemma 8.2 and Theorem A.3, and referring to (46), with

$$
\delta' = \sup_{\| (\mathbf{z}_1, \tilde{\mathbf{t}}_1) - (\mathbf{z}_2, \tilde{\mathbf{t}}_2) \| \le \frac{t}{(n\lambda h_n^p)^{1/2}}} d_{L,2}(\mathbf{z}_1, \mathbf{z}_2, \tilde{\mathbf{t}}_1, \tilde{\mathbf{t}}_2), \quad (47)
$$

we derive the following key results for $d=1$, for $t \ge c_5 \log n$,

$$
P^* \left\{ \omega_{\tilde{\xi}_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}\tilde{\mathbf{t}}); \frac{t}{(n\lambda h_n^p)^{1/2}} \right) \ge \frac{c_1 t}{(n\lambda h_n^p)^{1/4}} \right\}
$$
$$
\begin{align*}
&\le P^* \left\{ \omega(W_n; \rho_{L,2}; \delta') \ge \frac{c_1 t}{(n \lambda h_n^p)^{1/4}} \right\} \\
&\le P^* \left\{ \omega \left( W_n; \rho_{L,2}; \Psi(\lambda', \lambda_1, p, m, \theta, \delta) + \frac{c_1' t}{(n \lambda h_n^p)^{1/2}} \right) \ge \frac{c_1 t}{(n \lambda h_n^p)^{1/4}} \right\} \\
&\le P^* \left\{ \omega \left( W_n; \rho_{L,2}; \frac{c_1 t}{(n \lambda h_n^p)^{1/2}} \right) \ge \frac{c_1 t}{(n \lambda h_n^p)^{1/4}} \right\} \\
&\le c_3 \exp \left( -\frac{t}{c_4} \right). \tag{48}
\end{align*}
$$

Similarly, as demonstrated in (48), for $d > 2$, we obtain

$$
P^* \left\{ \omega_{\xi_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{I}}}) ; \frac{t}{(n \lambda h_n^p)^{1/2}} \right) \ge \frac{c_1 t (\log n)^{1/2}}{(n \lambda h_n^p)^{1/2}} \right\} \le c_3 \exp \left( -\frac{t}{c_4} \right). \quad (49)
$$

In an analogous manner, we derive the following results for $d > 2$

$$
P^* \left\{ \omega_{\xi_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{I}}}) ; \frac{t}{(n \lambda h_n^p)^{1/2}} \right) \ge \frac{c_1 t}{(n \lambda h_n^p)^{1/2}} \right\} \le c_3 \exp \left( -\frac{t}{c_4} \right). \quad (50)
$$

We proceed with the proof exclusively for the case $d > 2$, as the other cases follow similarly by substituting (48) or (49) in place of (50). To accomplish our objective, let us consider

$$
\begin{align*}
A &= \left\{ \Delta_{n,\lambda}(\mathfrak{B}_{\tilde{\mathfrak{I}}}) \ge (n\lambda h_n^p)^{-1} \left[ (d_1^2 + d_4) t + n \binom{n}{m}^{-1} \right] \right\}, \\
B &= \left\{ \omega_{\xi_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{I}}}); d_2(n\lambda h_n^p)^{-1/2} \|\xi_n\|_{\mathcal{V}} + d_3 \binom{n}{m}^{-1} \lambda h_n^{-p} \right) \ge d_1 t (n\lambda h_n^p)^{-1/2} \right\}, \\
C &= \left\{ \|\xi_n\|_{\mathcal{V}}^2 \ge t \right\}, \\
D &= \left\{ \|\xi_n\|_{\mathcal{V}} \ge (n\lambda h_n^p)^{1/2} \delta_0 - \binom{n}{m}^{-1} n^{1/2} (\lambda h_n^p)^{-1/2} \right\}.
\end{align*}
$$

Since (35) holds under $D^c$, Lemma 8.1 ensures that $D^c \cap B^c \cap C^c$ necessarily implies $A^c$, thus leading to

$$
\mathbb{P}(A) \le \mathbb{P}(B) + \mathbb{P}(C) + \mathbb{P}(D),
$$

i.e.,

$$
\begin{align}
& P^* \left( \Delta_{n,\lambda}(\mathfrak{B}_{\tilde{\mathfrak{I}}}) \ge (n\lambda h_n^p)^{-1} \left[ (d_1^2 + d_4) t + n \binom{n}{m}^{-1} \right] \right) \nonumber \\
&\le P^* \left( \omega_{\xi_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{I}}}); d_2(n\lambda h_n^p)^{-1/2} \|\xi_n\|_{\mathcal{V}} + d_3 \binom{n}{m}^{-1} \lambda h_n^{-p} \right) \ge d_1 t (n\lambda h_n^p)^{-1/2} \right) \nonumber \\
&\quad + P^* \left( \|\xi_n\|_{\mathcal{V}}^2 \ge t \right) + P^* \left( \|\xi_n\|_{\mathcal{V}} \ge (n\lambda h_n^p)^{1/2} \delta_0 - \binom{n}{m}^{-1} n^{1/2} (\lambda h_n^p)^{-1/2} \right). \tag{51}
\end{align}
$$

From (45), for

$$
0 < t < n \lambda h_n^p \delta_0^2 - 2n \binom{n}{m}^{-1} \delta_0 + n \binom{n}{m}^{-2} (\lambda h_n^p)^{-1},
$$

we obtain

$$
\begin{align*}
& P^* \left( \|\xi_n\|_{\mathcal{V}}^2 \ge t \right) + P \left( \|\xi_n\|_{\mathcal{V}} \ge (n \lambda h_n^p)^{1/2} \delta_0 - \binom{n}{m}^{-1} n^{1/2} (\lambda h_n^p)^{-1/2} \right) \\
\le{}& d_1 \exp(-t/d_2) + d_1 \exp \left( - \frac{n \lambda h_n^p \delta_0^2 - 2n \binom{n}{m}^{-1} \delta_0 + n \binom{n}{m}^{-2} (\lambda h_n^p)^{-1}}{d_2} \right) \\
\le{}& d_3 \exp \left( -\frac{t}{d_2} \right). \tag{52}
\end{align*}
$$

On the other hand, using

$$
\left\{ \omega_{\xi_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{I}}}); \frac{t^{1/2}}{(n \lambda h_n^p)^{1/2}} \right) \ge \frac{d_1 t}{n^{1/2}} \right\}^c \cap \left\{ \|\xi_n\|_{\mathcal{V}} \ge \frac{t^{1/2}}{d_2} \right\}^c \\
\subset \left\{ \omega_{\xi_n} \left( \tilde{U}_n^{-1}(\mathfrak{B}_{\tilde{\mathfrak{I}}}); d_2(n \lambda h_n^p)^{-1/2} \|\xi_n\|_{\mathcal{V}} + d_3 \binom{n}{m}^{-1} \lambda h_n^{-p} \right) \ge d_1 t (n \lambda h_n^p)^{-1/2} \right\},
$$
we have

$$
\begin{align*}
& P^* \left( \omega_{\xi_n} \left( \tilde{U}_n^{-1} (\mathfrak{B}_{\tilde{\mathfrak{t}}}) ; d_2 (n \lambda h_n^p)^{-1/2} \|\xi_n\|_{\mathcal{V}} + d_3 \binom{n}{m}^{-1} \lambda h_n^{-p} \right) \ge d_1 t (n \lambda h_n^p)^{-1/2} \right) \\
\le {} & P^* \left( \omega_{\xi_n} \left( \tilde{U}_n^{-1} (\mathfrak{B}_{\tilde{\mathfrak{t}}}) ; \frac{t^{1/2}}{(n \lambda h_n^p)^{1/2}} \right) \ge \frac{d_1 t}{n^{1/2}} \right) + P^* \left( \|\xi_n\|_{\mathcal{V}} \ge \frac{t^{1/2}}{d_2} \right) \\
\le {} & c_3 \exp \left( \frac{t}{c_4} \right). \tag{53}
\end{align*}
$$

Based on (51)–(53), we get that

$$
P^* \{ \Delta_{n,\lambda} (\mathfrak{B}_{\tilde{\mathfrak{t}}} | \tilde{\mathfrak{t}}) \ge (n \lambda h_n^p)^{-1} t \} \le c_3 \exp \left( \frac{-t}{c_4} \right), \quad (54)
$$

where

$$
c_5 \log n \le t \le n \lambda h_n^p \delta_0^2 - 2n \binom{n}{m}^{-1} \delta_0 + n \binom{n}{m}^{-2} (\lambda h_n^p)^{-1}.
$$

In order to establish the inequality for sufficiently large *t*, we observe that

$$
\begin{align*}
\Delta_{n,\lambda}(\mathfrak{B}_{\tilde{\mathfrak{t}}}|\tilde{\mathfrak{t}}) &= \left\| \mathbb{Q}_{\varphi,\lambda,\mathbb{H}_n}(\mathbf{u}|\tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi,\mathbb{H}}(\mathbf{u}|\tilde{\mathbf{t}}) \right. \\
&\qquad \left. + \operatorname{inv}((\mathbb{Q}_{\varphi,\mathbb{H}}^{-1})' \circ \mathbb{Q}_{\varphi,\mathbb{H}}(\mathbf{u}|\tilde{\mathbf{t}})) (\mathbb{Q}_{\varphi,\lambda,\mathbb{H}_n}^{-1} - \mathbb{Q}_{\varphi,\mathbb{H}}^{-1}) \circ \mathbb{Q}_{\varphi,\mathbb{H}}(\mathbf{u}|\tilde{\mathbf{t}}) \right\|_{\mathfrak{B}_{\tilde{\mathfrak{t}}}} \\
&\le \left\| \mathbb{Q}_{\varphi,\lambda,\mathbb{H}_n}(\mathbf{u}|\tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi,\mathbb{H}}(\mathbf{u}|\tilde{\mathbf{t}}) \right\|_{\mathfrak{B}_{\tilde{\mathfrak{t}}}} \\
&\qquad + \left\| \operatorname{inv}((\mathbb{Q}_{\varphi,\mathbb{H}}^{-1})' \circ \mathbb{Q}_{\varphi,\mathbb{H}}(\mathbf{u}|\tilde{\mathbf{t}})) (\mathbb{Q}_{\varphi,\lambda,\mathbb{H}_n}^{-1} - \mathbb{Q}_{\varphi,\mathbb{H}}^{-1}) \circ \mathbb{Q}_{\varphi,\mathbb{H}}(\mathbf{u}|\tilde{\mathbf{t}}) \right\|_{\mathfrak{B}_{\tilde{\mathfrak{t}}}} \\
&\le \left\| \mathbb{Q}_{\varphi,\lambda,\mathbb{H}_n}(\mathbf{u}|\tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi,\mathbb{H}}(\mathbf{u}|\tilde{\mathbf{t}}) \right\|_{\mathfrak{B}_{\tilde{\mathfrak{t}}}} + \mathfrak{D} \left\| \mathbb{Q}_{\varphi,\lambda,\mathbb{H}_n}^{-1} - \mathbb{Q}_{\varphi,\mathbb{H}}^{-1} \right\|_{\mathbb{Q}_{\varphi,\mathbb{H}}(\mathfrak{B}_{\tilde{\mathfrak{t}}})}. \tag{55}
\end{align*}
$$

From Lemma 8.1, noting that $\mathbb{Q}_{\varphi,\mathbb{H}}$ is also locally Lipschitz in $\mathcal{V}$, we have

$$
\begin{align*}
& \| \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}(\mathbf{u} | \tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi, \mathbb{H}}(\mathbf{u} | \tilde{\mathbf{t}}) \|_{\mathfrak{B}_{\tilde{\mathfrak{t}}}} \\
&\le \omega_{\mathbb{Q}_{\varphi, \mathbb{H}}} (\mathfrak{B}_{\tilde{\mathfrak{t}}}; \varrho (\tilde{U}_n, \tilde{U}_n^{-1}, \mathbb{Q}_{\varphi, \mathbb{H}}^{-1})) \\
&= \sup_{t \in \mathfrak{B}_{\tilde{\mathfrak{t}}}} \left\{ \left| \mathbb{Q}_{\varphi, \mathbb{H}}(s_1 | \tilde{\mathbf{t}}) - \mathbb{Q}_{\varphi, \mathbb{H}}(s_2 | \tilde{\mathbf{t}}) \right| : s_1, s_2 \in \mathbb{B}(t, \rho) \right\} \\
&\le \sup_{t \in \mathfrak{B}_{\tilde{\mathfrak{t}}}} \left\{ \mathcal{C} |s_1 - s_2| : |s_1 - s_2| \le \rho (\tilde{U}_n, \tilde{U}_n^{-1}, \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}) \right\} \\
&\le \mathcal{C} \cdot \rho (\tilde{U}_n, \tilde{U}_n^{-1}, \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}), \tag{56}
\end{align*}
$$

where

$$
\varrho(\tilde{U}_n, \tilde{U}_n^{-1}, \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}) = \delta(\tilde{U}_n) + \eta(\tilde{U}_n, \tilde{U}_n^{-1}) + \|\mathbb{E}[\tilde{U}_n] - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}\|_{\mathcal{V}}.
$$

Furthermore, we can see that

$$
\begin{align*}
\| \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}^{-1} - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1} \|_{\mathbb{Q}_{\varphi, \mathbb{H}}(\mathfrak{B}_{\tilde{\mathfrak{t}}})} &\le \tilde{\mathfrak{D}} \| \mathbb{Q}_{\varphi, \lambda, \mathbb{H}_n}^{-1} - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1} \|_{\mathcal{V}} \\
&= \tilde{\mathfrak{D}} \left\| \frac{U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}})}{f_n(\lambda, \tilde{\mathbf{t}})} + \frac{U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}})}{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} - \frac{U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}})}{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1} \right\|_{\mathcal{V}} \\
&\le \tilde{\mathfrak{D}} \left\| \frac{U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}})}{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1} \right\|_{\mathcal{V}} + \tilde{\mathfrak{D}} \left\| U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) \left( \frac{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}})) - f_n(\lambda, \tilde{\mathbf{t}})}{f_n(\lambda, \tilde{\mathbf{t}}) \cdot \mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} \right) \right\|_{\mathcal{V}} \\
&\le \tilde{\mathfrak{D}} \left( \|\tilde{U}_n - \mathbb{E}[\tilde{U}_n]\|_{\mathcal{V}} + \|\mathbb{E}[\tilde{U}_n] - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}\|_{\mathcal{V}} \right) \\
&\qquad + \tilde{\mathfrak{D}} \left\| U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) \left( \frac{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}})) - f_n(\lambda, \tilde{\mathbf{t}})}{f_n(\lambda, \tilde{\mathbf{t}}) \cdot \mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} \right) \right\|_{\mathcal{V}}. \tag{57}
\end{align*}
$$

From (36), with $S \equiv 1$, we get that

$$
\left\| U_n(\varphi, \mathbf{z}, \lambda, \tilde{\mathbf{t}}) \left( \frac{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}})) - f_n(\lambda, \tilde{\mathbf{t}})}{f_n(\lambda, \tilde{\mathbf{t}}) \cdot \mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} \right) \right\|_{\mathcal{V}} \le \mathcal{C} (n \lambda_1 h_n^p)^{-1/2} \|\xi_n\|_{\mathcal{V}}. \quad (58)
$$

According to Theorem 4.3 and using (36), (37), (55), (56) and (58), we obtain,

$$
\Delta_{n,\lambda}(\mathfrak{B}_{\tilde{\mathfrak{t}}}|\tilde{\mathfrak{t}}) \le \mathcal{C}_1 \cdot \rho(\tilde{U}_n, \tilde{U}_n^{-1}, \mathbb{Q}_{\varphi,\mathbb{H}}^{-1}) + \mathcal{C}_2 \|\mathbb{Q}_{\varphi,\lambda,\mathbb{H}_n}^{-1} - \mathbb{Q}_{\varphi,\mathbb{H}}^{-1}\|_{\mathcal{V}}
$$
$$
\begin{align}
&\le \mathcal{C}_1 \cdot \rho (\tilde{U}_n, \tilde{U}_n^{-1}, \mathbb{Q}_{\phi, \mathbb{H}}^{-1}) + \mathcal{C}_2 \left( \left\| \tilde{U}_n - \mathbb{E}[\tilde{U}_n] \right\|_{\mathbb{V}} + \left\| \mathbb{E}[\tilde{U}_n] - \mathbb{Q}_{\phi, \mathbb{H}}^{-1} \right\|_{\mathbb{V}} \right) \nonumber \\
&\quad + \left\| U_n (\varphi, z, \lambda, \tilde{\mathbf{t}}) \left( \frac{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}})) - f_n(\lambda, \tilde{\mathbf{t}})}{f_n(\lambda, \tilde{\mathbf{t}}) \cdot \mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} \right) \right\|_{\mathbb{V}} \nonumber \\
&= \mathcal{C}_1 (\delta(\tilde{U}_n) + \eta(\tilde{U}_n, \tilde{U}_n^{-1}) + \left\| \mathbb{E}[\tilde{U}_n] - \mathbb{Q}_{\phi, \mathbb{H}}^{-1} \right\|_{\mathbb{V}}) + \mathcal{C}_2 (\delta(\tilde{U}_n) + \left\| \mathbb{E}[\tilde{U}_n] - \mathbb{Q}_{\phi, \mathbb{H}}^{-1} \right\|_{\mathbb{V}}) \nonumber \\
&\quad + \left\| U_n (\varphi, z, \lambda, \tilde{\mathbf{t}}) \left( \frac{\mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}})) - f_n(\lambda, \tilde{\mathbf{t}})}{f_n(\lambda, \tilde{\mathbf{t}}) \cdot \mathbb{E}(f_n(\lambda, \tilde{\mathbf{t}}))} \right) \right\|_{\mathbb{V}} \nonumber \\
&\le \tilde{\mathcal{C}} \delta (\tilde{U}_n) + \tilde{\mathcal{C}}' \lambda_1^{-1} h_n^{-p} \left( \frac{n}{m} \right)^{-1} \nonumber \\
&\le \mathcal{C} \left( (n \lambda_1 h_n^p)^{-1/2} \|\xi_n\|_{\mathbb{V}} + \lambda_1^{-1} h_n^{-p} \left( \frac{n}{m} \right)^{-1} \right). \tag{59}
\end{align}
$$

Now, for $t \ge \mathcal{C}_n \lambda_1 h_n^p$, we can apply Theorem A.3 to obtain

$$
\begin{align}
\mathbb{P}^* \left\{ \|\xi_n\|_{\mathbb{V}} \ge tn^{-1/2} (\lambda_1 h_n^p)^{-1/2} \right\} &\le \mathcal{C}_3 \exp \left( - \frac{t^2}{\mathcal{C}_4 n \lambda_1 h_n^p} \right) \nonumber \\
&\le \mathcal{C}_3 \exp \left( - \frac{\mathcal{C}_n \lambda_1 h_n^p t}{\mathcal{C}_4 n \lambda_1 h_n^p} \right) \nonumber \\
&\le \mathcal{C}_3 \exp \left( \frac{-t}{\mathcal{C}_5} \right). \tag{60}
\end{align}
$$

Combining (59) and (60), we obtain, for $t \ge \mathcal{C}_1 n \lambda_1 h_n^p$,

$$
\begin{align}
& \mathbb{P}^* \left\{ \Delta_{n,\lambda} (\mathfrak{W}_{\tilde{\mathbf{t}}} | \tilde{\mathbf{t}}) \ge tn^{-1} (\lambda_1 h_n^p)^{-1} \right\} \nonumber \\
&\le \mathbb{P}^* \left\{ \mathcal{C}_5 \left( (n\lambda_1 h_n^p)^{-1/2} \|\xi_n\|_{\mathbb{V}} + \lambda_1^{-1} h_n^{-p} \left(\frac{n}{m}\right)^{-1} \right) \ge tn^{-1} (\lambda_1 h_n^p)^{-1} \right\} \nonumber \\
&\le \mathbb{P}^* \left\{ \mathcal{C}_5 \left( \|\xi_n\|_{\mathbb{V}} + n^{1/2} (\lambda_1 h_n^p)^{-1/2} \left(\frac{n}{m}\right)^{-1} \right) \ge tn^{-1/2} (\lambda_1 h_n^p)^{-1/2} \right\} \nonumber \\
&\le \mathbb{P}^* \left\{ \|\xi_n\|_{\mathbb{V}} \ge \mathcal{C}_2 tn^{-1/2} (\lambda_1 h_n^p)^{-1/2} \right\} \nonumber \\
&\le \mathcal{C}_3 \exp \left( \frac{-t}{\mathcal{C}_4} \right). \tag{61}
\end{align}
$$

It follows from (54) and (61), when $d > 2$,

$$
\mathbb{P}^* \left\{ \Delta_{n,\lambda} (\mathfrak{W}_{\tilde{\mathbf{t}}} | \tilde{\mathbf{t}}) \ge (n\lambda h_n^p)^{-1} t \right\} \le \beta_1 \exp \left( \frac{-t}{\beta_2} \right), \quad t > \beta_3 \log n. \tag{62}
$$

Similar arguments lead to the desired inequalities for $d = 1$ and $d = 2$.

**Proof of Theorem 4.3.** By following the same procedures as in [100], we remark that

$$
\begin{align}
& \left| \mathbb{E} U_n(\varphi, z, \lambda, \tilde{\mathbf{t}}) - \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(\mathbf{u} | \tilde{\mathbf{t}}) \right| \nonumber \\
&= \left| \frac{1}{\lambda^m h_n^{pm}} \int_{\mathbb{R}^{pm} \times \mathbb{R}^d} S(\tilde{z} - (\mathbf{w}_1, \dots, \mathbf{w}_d)) \prod_{j=1}^m K \left( \frac{\mathbf{t}_j - \mathbf{v}_j}{\lambda h_n} \right) f_{XY}(\mathbf{w}_j, \mathbf{v}_j) d\mathbf{w}_j d\mathbf{v}_j - \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(\mathbf{u} | \tilde{\mathbf{t}}) \right| \nonumber \\
&= \left| \frac{1}{\lambda^m h_n^{pm}} \int_{\mathbb{R}^{pm}} \prod_{j=1}^m K \left( \frac{\mathbf{t}_j - \mathbf{v}_j}{\lambda h_n} \right) \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbf{z} | \mathbf{v}_j) f_X(\mathbf{v}_j) d\mathbf{v}_j - \tilde{\mathbb{Q}}_{\varphi, \mathbb{H}}^{-1}(\mathbf{u} | \tilde{\mathbf{t}}) \right| \nonumber \\
&= \left| \int_{\mathbb{R}^{pm}} \prod_{j=1}^m K(\mathbf{s}_j) \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbf{z} | \mathbf{t}_j - \lambda h_n \mathbf{s}_j) \prod_{j=1}^m f_X(\mathbf{t}_j - \lambda h_n \mathbf{s}_j) d\mathbf{s}_j - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbf{z} | \tilde{\mathbf{t}}) \prod_{j=1}^m f_X(\mathbf{t}_j) \right| \nonumber \\
&= \left| \int_{\mathbb{R}^{pm}} \prod_{j=1}^m K(\mathbf{s}_j) \left( \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbf{z} | \mathbf{t}_j - \lambda h_n \mathbf{s}_j) \cdot \prod_{j=1}^m f_X(\mathbf{t}_j - \lambda h_n \mathbf{s}_j) - \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbf{z} | \tilde{\mathbf{t}}) \cdot \prod_{j=1}^m f_X(\mathbf{t}_j) \right) d\mathbf{s}_j \right| \nonumber \\
&= \left| \int_{\mathbb{R}^{pm}} \prod_{j=1}^m K(\mathbf{s}_j) \cdot \mathbb{Q}_{\varphi, \mathbb{H}}^{-1}(\mathbf{z} | \mathbf{t}_j - \lambda h_n \mathbf{s}_j) (\Theta_{\mathrm{t},s}(1) - \Theta_{\mathrm{t},s}(0)) d\mathbf{s}_j \right|, \tag{63}
\end{align}
$$

where $\Theta_{\mathrm{t},s}(\beta) := \prod_{j=1}^m f_X(\mathbf{t}_j - \beta \lambda h_n \mathbf{s}_j) - f_X(\mathbf{t}_j)$ for $\beta \in [-1, 1]$. Then, we have

$$
\Theta_{\mathrm{t},s}^{(\alpha)}(0) = \sum_{k_1+\cdots+k_m=\alpha} \left( \begin{array}{c} \alpha \\ k_1:m \end{array} \right) \prod_{j=1}^{m} \frac{\partial^{k_j} (f_X(\mathbf{t}_j - \beta \lambda h_n \mathbf{s}_j))}{\partial \beta^{k_j}}(0)
$$
$$ = \sum_{k_1+\cdots+k_m=\alpha} \left( \binom{\alpha}{k_1:m} \right) \prod_{j=1}^{m} \sum_{l_1, \ldots, l_{k_j}=1}^{m} (-1)^{k_j} (\lambda h_n)^{k_j} \mathbf{s}_{l_1} \cdots \mathbf{s}_{l_{k_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{l_1} \cdots \partial \mathbf{t}_{l_{k_j}}} (\mathbf{t}_j - \beta_{\mathrm{t},s} \lambda h_n \mathbf{s}_j), $$

where

$$ \left( \begin{array}{c} \alpha \\ k_1:m \end{array} \right) := \alpha! / \left( \prod_{j=1}^{m} (k_j!) \right), $$

is the multinomial coefficient. In the same manner, denote

$$ \tilde{\mathfrak{R}}_{\mathrm{t},s}(\beta) := \prod_{j=1}^{m} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbf{z} \mid \mathbf{t}_j - \lambda h_n \mathbf{s}_j), $$

and we have

$$ \begin{aligned} \tilde{\mathfrak{R}}^{(\alpha)}_{\mathrm{t},s}(0) &= \sum_{k_1+\cdots+k_m=\alpha} \left( \binom{\alpha}{k_1:m} \right) \prod_{j=1}^{m} \frac{\partial^{k_j} \left( \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbf{z} \mid \mathbf{t}_j - \beta \lambda h_n \mathbf{s}_j) \right)}{\partial \beta^{k_j}} (0) \\ &= \sum_{k_1+\cdots+k_m=\alpha} \left( \binom{\alpha}{k_1:m} \right) \prod_{j=1}^{m} \sum_{l_1, \ldots, l_{k_j}=1}^{m} (-1)^{k_j} (\lambda h_n)^{k_j} \mathbf{s}_{l_1} \cdots \mathbf{s}_{l_{k_j}} \\ & \quad \times \frac{\partial^{k_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{l_1} \cdots \partial \mathbf{t}_{l_{k_j}}} (\mathbf{z} \mid \mathbf{t}_j - \beta_{\mathrm{t},s} \lambda h_n \mathbf{s}_j). \end{aligned} $$

Let $\mathfrak{N}(\beta) := \tilde{\mathfrak{R}}_{\mathrm{t},s}(\beta)\Theta_{\mathrm{t},s}(\beta)$. By using Leibniz's rule, we obtain

$$ \begin{aligned} \mathfrak{N}^{(l)}(0) &= \sum_{\ell=0}^{l} \binom{l}{\ell} \tilde{\mathfrak{R}}_{\mathrm{t},s}^{(\ell)}(0) \Theta_{\mathrm{t},s}^{(l-\ell)}(0) \\ &= \sum_{\ell=0}^{l} \binom{l}{\ell} \sum_{k_1+\cdots+k_m=\ell} \sum_{l_1+\cdots+l_m=l-\ell} \binom{\ell}{k_1:m} \binom{l-\ell}{l_1:m} \\ & \quad \times \prod_{j=1}^{m} \sum_{l_1, \ldots, l_{k_j}=1}^{m} \sum_{\tilde{l}_1, \ldots, \tilde{l}_{k_j}=1}^{m} (-1)^{k_j} (-1)^{l} (\lambda h_n)^{k_j} (\lambda h_n)^{l} \mathbf{s}_{l_1} \cdots \mathbf{s}_{l_{k_j}} \tilde{\mathbf{s}}_{\tilde{l}_1} \cdots \tilde{\mathbf{s}}_{\tilde{l}_{k_j}} \\ & \quad \times \frac{\partial^{k_j} f}{\partial \mathbf{t}_{l_1} \cdots \partial \mathbf{t}_{l_{k_j}}} (\mathbf{t}_j - \beta_{\mathrm{t},s} \lambda h_n \mathbf{s}_j) \frac{\partial^{l} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{l_1} \cdots \partial \mathbf{t}_{l_{k_j}}} (\mathbf{z} \mid \tilde{\mathbf{t}} - \mathbf{s}_{\mathrm{t},s} \lambda h_n \tilde{\mathbf{s}}) \\ &= \sum_{\ell=0}^{l} \binom{l}{\ell} (-1)^{l} (\lambda h_n)^{l} \sum_{k_1+\cdots+k_m=\ell} \sum_{l_1+\cdots+l_m=l-\ell} \binom{\ell}{k_1:m} \binom{l-\ell}{l_1:m} \\ & \quad \times \prod_{j=1}^{m} \sum_{l_1, \ldots, l_{k_j}=1}^{m} \sum_{\tilde{l}_1, \ldots, \tilde{l}_{k_j}=1}^{m} \mathbf{s}_{l_1} \cdots \mathbf{s}_{l_{k_j}} \tilde{\mathbf{s}}_{\tilde{l}_1} \cdots \tilde{\mathbf{s}}_{\tilde{l}_{k_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{l_1} \cdots \partial \mathbf{t}_{l_{k_j}}} (\mathbf{t}_j - \beta_{\mathrm{t},s} \lambda h_n \mathbf{s}_j) \\ & \quad \times \frac{\partial^{l} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{l_1} \cdots \partial \mathbf{t}_{l_{k_j}}} (\mathbf{z} \mid \tilde{\mathbf{t}} - \beta_{\mathrm{t},s} \lambda h_n \tilde{\mathbf{s}}). \end{aligned} $$

Therefore, if the function $\mathfrak{N}(\cdot)$ is $r$-times differentiable, applying a Taylor-Lagrange expansion and utilizing (iii) in Assumption 2, it follows from (63) that:

$$ \begin{aligned} & \left| \int_{\mathbb{R}^{pm}} \prod_{j=1}^{m} K(\mathbf{s}_j) \cdot \mathbb{Q}_{\phi, \mathbb{H}}^{-1}(\mathbf{z} \mid \mathbf{t}_j - \lambda h_n \mathbf{s}_j) (\Theta_{\mathrm{t},s}(1) - \Theta_{\mathrm{t},s}(0)) \, \mathrm{d}\mathbf{s}_j \right| \\ &= \left| \int_{\mathbb{R}^{pm}} \left( \sum_{l=1}^{r-1} \frac{1}{l!} \mathfrak{N}^{(l)}(0) + \frac{1}{r!} \mathfrak{N}^{(r)}(\beta) \right) \prod_{j=1}^{m} K(\mathbf{s}_j) \, \mathrm{d}\mathbf{s}_j \right| \\ &\le \left| \int_{\mathbb{R}^{pm}} \sum_{l=1}^{r-1} \frac{1}{l!} \mathfrak{N}^{(l)}(0) \prod_{j=1}^{m} K(\mathbf{s}_j) \, \mathrm{d}\mathbf{s}_j \right| + \left| \int_{\mathbb{R}^{pm}} \frac{1}{r!} \mathfrak{N}^{(r)}(\beta) \prod_{j=1}^{m} K(\mathbf{s}_j) \, \mathrm{d}\mathbf{s}_j \right| \\ &\le (\lambda h_n)^r \left| \int_{\mathbb{R}^{pm}} \sum_{\ell=0}^{r-1} \left( \frac{\mathbf{r}}{\ell} \right) \sum_{l=1}^{r-1} \frac{1}{l!} (-1)^r \sum_{k_1+\cdots+k_m=\ell} \sum_{\mathbf{r}_1+\cdots+\mathbf{r}_m=\mathbf{r}-\ell} \binom{\ell}{k_1:m} \binom{\mathbf{r}-\ell}{l_1:m} \right. \\ & \qquad \left. \times \prod_{j=1}^{m} \sum_{l_1, \ldots, l_{k_j}=1}^{m} \sum_{\tilde{l}_1, \ldots, \tilde{l}_{k_j}=1}^{m} \mathbf{s}_{l_1} \cdots \mathbf{s}_{l_{k_j}} \tilde{\mathbf{s}}_{\tilde{l}_1} \cdots \tilde{\mathbf{s}}_{\tilde{l}_{k_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{l_1} \cdots \partial \mathbf{t}_{l_{k_j}}} (\mathbf{t}_j - \beta_{\mathrm{t},s} \lambda h_n \mathbf{s}_j) \right| \end{aligned} $$
$$
\begin{align*}
& \times \frac{\partial^{l_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{l_j}}} (\mathbf{z} | \tilde{\mathbf{t}} - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \tilde{\mathbf{s}}) \prod_{j=1}^{m} K(s_j) \, ds_j \\
& + \frac{(\lambda h_n)^r}{r!} \left| \int_{\mathbb{R}^{pm}} \sum_{\ell=0}^{r} \left(\frac{\mathbf{r}}{\ell}\right)^r \sum_{l=1}^{r} (-1)^r \sum_{k_1+\cdots+k_m=\ell} \sum_{\mathbf{r}_1+\cdots+\mathbf{r}_m=\mathbf{r}-\ell} \left(\begin{smallmatrix} \ell \\ k_1:m \end{smallmatrix}\right) \left(\begin{smallmatrix} \mathbf{r}-\ell \\ l_1:m \end{smallmatrix}\right) \right. \\
& \qquad \left. \times \prod_{j=1}^{m} \sum_{i_1, \ldots, i_{k_j}=1}^{m} \sum_{\tilde{\mathbf{t}}_1, \ldots, \tilde{\mathbf{t}}_{i_j}=1}^{m} \mathbf{s}_{i_1} \cdots \mathbf{s}_{i_{k_j}} \mathbf{s}_{\tilde{\mathbf{t}}_1} \cdots \mathbf{s}_{\tilde{\mathbf{t}}_{i_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{k_j}}} (\mathbf{t}_j - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \mathbf{s}_j) \right. \\
& \qquad \left. \times \frac{\partial^{l_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{l_j}}} (\mathbf{z} | \tilde{\mathbf{t}} - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \tilde{\mathbf{s}}) \prod_{j=1}^{m} K(s_j) \, ds_j \right| \\
& \le \frac{(\lambda h_n)^r}{r!} \left| \int_{\mathbb{R}^{pm}} \sum_{\ell=0}^{r} \left(\frac{\mathbf{r}}{\ell}\right)^r \sum_{l=1}^{r} (-1)^r \sum_{k_1+\cdots+k_m=\ell} \sum_{\mathbf{r}_1+\cdots+\mathbf{r}_m=\mathbf{r}-\ell} \left(\begin{smallmatrix} \ell \\ k_1:m \end{smallmatrix}\right) \left(\begin{smallmatrix} \mathbf{r}-\ell \\ l_1:m \end{smallmatrix}\right) \right. \\
& \qquad \left. \times \prod_{j=1}^{m} \sum_{i_1, \ldots, i_{k_j}=1}^{m} \sum_{\tilde{\mathbf{t}}_1, \ldots, \tilde{\mathbf{t}}_{i_j}=1}^{m} \mathbf{s}_{i_1} \cdots \mathbf{s}_{i_{k_j}} \mathbf{s}_{\tilde{\mathbf{t}}_1} \cdots \mathbf{s}_{\tilde{\mathbf{t}}_{i_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{k_j}}} (\mathbf{t}_j - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \mathbf{s}_j) \right. \\
& \qquad \left. \times \frac{\partial^{l_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{l_j}}} (\mathbf{z} | \tilde{\mathbf{t}} - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \tilde{\mathbf{s}}) \left|_{s_j=p(\mathbf{t}_j, \lambda h_n s_j)} \prod_{j=1}^{m} K(s_j) \, ds_j \right| \right| \\
& \le \frac{(\lambda h_n)^r}{r!} \left| \int_{\mathbb{R}^{pm}} \sum_{\ell=0}^{r} \left(\frac{\mathbf{r}}{\ell}\right)^r \sum_{l=1}^{r} (-1)^r \sum_{k_1+\cdots+k_m=\ell} \sum_{\mathbf{r}_1+\cdots+\mathbf{r}_m=\mathbf{r}-\ell} \left(\begin{smallmatrix} \ell \\ k_1:m \end{smallmatrix}\right) \left(\begin{smallmatrix} \mathbf{r}-\ell \\ l_1:m \end{smallmatrix}\right) \right. \\
& \qquad \left. \times \prod_{j=1}^{m} \sum_{i_1, \ldots, i_{k_j}=1}^{m} \sum_{\tilde{\mathbf{t}}_1, \ldots, \tilde{\mathbf{t}}_{i_j}=1}^{m} \mathbf{s}_{i_1} \cdots \mathbf{s}_{i_{k_j}} \mathbf{s}_{\tilde{\mathbf{t}}_1} \cdots \mathbf{s}_{\tilde{\mathbf{t}}_{i_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{k_j}}} (\mathbf{t}_j - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \mathbf{s}_j) \right. \\
& \qquad \left. \times \frac{\partial^{l_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{l_j}}} (\mathbf{z} | \tilde{\mathbf{t}} - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \tilde{\mathbf{s}}) \left|_{s_j=p(\mathbf{t}_j, \lambda h_n s_j)} \prod_{j=1}^{m} K(s_j) \, ds_j \right| \right| \\
& \le \frac{(\lambda h_n)^r}{r!} \int_{\mathbb{R}^{pm}} \sum_{\ell=0}^{r} \left(\frac{\mathbf{r}}{\ell}\right)^r \sum_{l=1}^{r} (-1)^r \sum_{k_1+\cdots+k_m=\ell} \sum_{\mathbf{r}_1+\cdots+\mathbf{r}_m=\mathbf{r}-\ell} \left(\begin{smallmatrix} \ell \\ k_1:m \end{smallmatrix}\right) \left(\begin{smallmatrix} \mathbf{r}-\ell \\ l_1:m \end{smallmatrix}\right) \right. \\
& \qquad \left. \times \prod_{j=1}^{m} \sum_{i_1, \ldots, i_{k_j}=1}^{m} \sum_{\tilde{\mathbf{t}}_1, \ldots, \tilde{\mathbf{t}}_{i_j}=1}^{m} \mathbf{s}_{i_1} \cdots \mathbf{s}_{i_{k_j}} \mathbf{s}_{\tilde{\mathbf{t}}_1} \cdots \mathbf{s}_{\tilde{\mathbf{t}}_{i_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{k_j}}} (\mathbf{t}_j - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \mathbf{s}_j) \right. \\
& \qquad \left. \times \frac{\partial^{l_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{l_j}}} (\mathbf{z} | \tilde{\mathbf{t}} - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \tilde{\mathbf{s}}) \left| \prod_{j=1}^{m} K(s_j) \, ds_j \right| \right| \\
& \le \frac{(\lambda h_n)^r}{r!} \sum_{\ell=0}^{r} \left(\frac{\mathbf{r}}{\ell}\right)^r \sum_{l=1}^{r} (-1)^r \sum_{k_1+\cdots+k_m=\ell} \sum_{\mathbf{r}_1+\cdots+\mathbf{r}_m=\mathbf{r}-\ell} \left(\begin{smallmatrix} \ell \\ k_1:m \end{smallmatrix}\right) \left(\begin{smallmatrix} \mathbf{r}-\ell \\ l_1:m \end{smallmatrix}\right) \times \prod_{j=1}^{m} \sum_{i_1, \ldots, i_{k_j}=1}^{m} \sum_{\tilde{\mathbf{t}}_1, \ldots, \tilde{\mathbf{t}}_{i_j}=1}^{m} \mathbf{s}_{i_1} \cdots \mathbf{s}_{i_{k_j}} \mathbf{s}_{\tilde{\mathbf{t}}_1} \cdots \mathbf{s}_{\tilde{\mathbf{t}}_{i_j}} \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{k_j}}} (\mathbf{t}_j - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \mathbf{s}_j) \\
& \qquad \times \frac{\partial^{l_j} \mathbb{Q}_{\phi, \mathbb{H}}^{-1}}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{l_j}}} (\mathbf{z} | \tilde{\mathbf{t}} - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \tilde{\mathbf{s}}) \left| \left( \int_{\mathbb{R}^{pm}} \left| \frac{\partial^{k_j} f_X}{\partial \mathbf{t}_{i_1} \cdots \partial \mathbf{t}_{i_{k_j}}} (\mathbf{t}_j - \beta_{\mathbf{t}, \mathbf{s}} \lambda h_n \mathbf{s}_j) \right| \right)^{1/2} \left( \int_{\mathbb{R}^{pm}} \prod_{j=1}^{m} |K(s_j)|^2 ds_j \right) \right|^{1/2} \right) \\
& = O(h_n^r). \tag{64}
\end{align*}
$$

Hence, the proof is completed. ☐

Funding

This work was supported by ongoing institutional funding. No additional grants were obtained to carry out or direct this particular research.
## 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

We begin by recalling a key lemma, adapted from Proposition 4 in [101], which provides a uniform exponential bound for the nonlinear portions of Hoeffding-type decompositions, as represented in (28), in the setting of a uniformly bounded VC family of functions.

**Lemma A.1.** Let $\mathcal{L}$ be a uniformly bounded VC class of real-valued functions. Then there is a positive constant $c$ such that, for all $t > 0$,

$$ \mathbb{P}^* \left( \sqrt{n} \left\| \sum_{k=2}^{m} \binom{m}{k} U_n (\pi_{k,m} f) \right\|_{\mathcal{L}} \ge t \right) \le 2 \exp \left( -\frac{t \sqrt{n}}{c} \right). $$

As an additional preliminary step, let us consider a family $\mathcal{G}$ of real-valued functions, as defined in (10), for which $\mathbb{H}(z | \tilde{t})$ denotes the associated cumulative distribution function (cdf), and $\mathbb{H}_n(z | \tilde{t})$ its empirical counterpart. We define the corresponding conditional U-process indexed by $f \circ \varphi$, where $f$ ranges over $\mathcal{G}$, via

$$ W_n(f) := \sqrt{n}(\mathbb{P}_n - \mathbb{P})(f), \quad f \in \mathcal{G}. $$

In addition, we set

$$ \sigma^2 = \sigma^2(\mathcal{G}) := \sup_{f \in \mathcal{G}} \mathrm{Var}(\pi_{1,m} f \circ \varphi(\mathbf{X}_1, \mathbf{Y}_1)). $$

**Theorem A.2.** Assume $\mathcal{G}$ is a VC class. Then there exist constants $d_1, d_2 > 0$ such that, for all $t > 0$,

$$ \mathbb{P}^* \left( \sup_{f \in \mathcal{G}} |W_n(f)| \ge t \right) \le d_1 \exp \left( -\frac{t^2}{d_2} \right). \qquad (65) $$

In addition, there exist constants $d_3, \dots, d_8 > 0$ such that, whenever

$$ t > (d_3 n^{-1/2} \log n) \lor (d_4 \sigma \sqrt{\log(1/\sigma)}), $$

we have

$$ \mathbb{P}^* \left( \sup_{f \in \mathcal{G}} |W_n(f)| \ge t \right) \le d_5 \exp \left( -\frac{t^2}{d_6 \sigma^2} \right) \lor d_7 \exp \left( -\frac{t n^{1/2}}{d_8} \right). $$

**Theorem A.3.** Consider $\{f-g: f,g \in \mathcal{G}\}$, where $\mathcal{G}$ is a VC class generated by uniformly bounded functions. Fix any $\alpha \in (0,1]$. Then there are constants $c_1, \dots, c_6$ such that

$$ \mathbb{P}^* \left( \omega(W_n; \rho_{L,2}; t^{1/2}/n^{\alpha/2}) \ge \frac{t}{\sqrt{n^\alpha \lambda h_n^{\alpha p}}} \right) \le c_1 \exp \left( -\frac{t}{c_2} \right), \quad t \ge c_3 \log n, $$

and

$$ \mathbb{P}^* \left( \omega(W_n; \rho_{L,2}; t^{1/2}/(n^{1/2} \sqrt{\log n})) \ge \frac{t}{\sqrt{\lambda h_n^{\alpha p} n \log n}} \right) \le c_4 \exp \left( -\frac{t}{c_5} \right), \quad t \ge c_6 \log n. $$

**Lemma A.4.** Let $\mathcal{G}$ be a uniformly bounded VC class. Then there exist constants $C_3, C_4 > 0$ such that, for all $s > 0$ and $n \ge m$,

$$ \mathbb{E} \left\{ \exp \left( s \| \mathbb{P}_n - \mathbb{P} \|_{\mathcal{G}} \right) \right\} \le \left( 1 + C_3 s [n/m]^{-1/2} \right) \exp \left( \frac{C_4 s^2}{4[n/m]} \right). \qquad (66) $$

**Proof of Theorem A.2.** We use Lemma A.4 in conjunction with the fundamental relation

$$ \mathbb{P} (\|\mathbb{P}_n - \mathbb{P}\|_{\mathcal{G}} \ge t) \le \exp(-st) \cdot \mathbb{E} \{\exp(s \|\mathbb{P}_n - \mathbb{P}\|_{\mathcal{G}})\} $$

and take $s = 2[n/m]t/C_4$. Then

$$ \mathbb{P} (\|W_n\|_{\mathcal{G}} \ge t) = \mathbb{P} (\|\mathbb{P}_n - \mathbb{P}\|_{\mathcal{G}} \ge t n^{-1/2}) $$
$$
\begin{align*}
&\leq \exp(-st) \cdot \mathbb{E} \{ \exp(s \|P_n - P\|_G) \} \\
&\leq (1 + C_3 \left[ \frac{[n/m]}{n} \right)^{1/2t} ) \exp(-[n/m]t^2/C_4n) \\
&\leq c_1 \exp(-t^2/c_2),
\end{align*}
$$

for suitable $c_1$ and $c_2$, proving (66). Now, by applying The Hoeffding decomposition, we get that

$$
\begin{align*}
W_n (G_{\phi,z,\lambda,\bar{\imath}}) &= \sqrt{n} (P_n - P) (G_{\phi,z,\lambda,\bar{\imath}}) \\
&= \sqrt{n} \{ U_n^{(m)} (G_{\phi,z,\lambda,\bar{\imath}}) - \mathbb{E} U_n^{(m)} (G_{\phi,z,\lambda,\bar{\imath}}) \} \\
&= \sqrt{n} \sum_{k=1}^{m} \binom{m}{k} U_n^{(k)} (\pi_k G_{\phi,z,\lambda,\bar{\imath}}) \\
&= \sqrt{n} (m U_n^{(1)} (\pi_1 G_{\phi,z,\lambda,\bar{\imath}})) + \sqrt{n} \sum_{k=2}^{m} \binom{m}{k} U_n^{(k)} (\pi_k G_{\phi,z,\lambda,\bar{\imath}}).
\end{align*}
$$

Therefore, by Lemma A.1 and Lemma A.2 in [42], we obtain that

$$
\begin{align*}
\mathbb{P} \left( \sup_{\mathcal{G}} W_n (G_{\phi,z,\lambda,\bar{\imath}}) \right) &\le \mathbb{P} \left( \left\| n^{-1/2} \sum_{i=1}^n \pi_1 G_{\phi,z,\lambda,\bar{\imath}} (\mathbf{X}_i, \mathbf{Y}_i) \right\|_{\mathcal{G}} \ge t/m \right) + \exp(-tn^{1/2}/c) \\
&\le \mathbb{P} \left( \left\| Z_n (\pi_1 G_{\phi,z,\lambda,\bar{\imath}}) \right\|_{\mathcal{G}} \ge t/m \right) + \exp(-tn^{1/2}/c) \\
&\le \max \left\{ c_5 \exp(-t^2/c_6\sigma^2), \exp(-tn^{1/2}/c_7) \right\} + 2 \exp(-tn^{1/2}/c) \\
&\le \max \left\{ d_5 \exp(-t^2/d_6\sigma^2), d_7 \exp(-tn^{1/2}/d_8) \right\} \tag{67}
\end{align*}
$$

for suitable choices of constants.

**Proof of Theorem A.3.** Without loss of generality, we assume that $\mathbb{E}G_{\phi,z,\lambda,\bar{\imath}} = 0$. Then, it follows that

$$
\begin{align*}
\mathrm{Var} (\pi_k G_{\phi,z,\lambda,\bar{\imath}} (\mathbf{X}, \mathbf{Y})) &= \mathbb{E} \left( [\pi_k G_{\phi,z,\lambda,\bar{\imath}} (\mathbf{X}, \mathbf{Y})]^2 \right) \\
&= \mathbb{E} \left\{ \left[ \mathbb{E} G_{\phi,z,\lambda,\bar{\imath}} (\mathbf{X}, \mathbf{Y}) \mid (\mathbf{X}_1, \mathbf{Y}_1) = (\mathbf{X}, \mathbf{Y}) \right]^2 \right\} \\
&\le \mathbb{E} \left\{ \left[ \mathbb{E} G_{\phi,z,\lambda,\bar{\imath}}^2 (\mathbf{X}, \mathbf{Y}) \mid (\mathbf{X}_1, \mathbf{Y}_1) = (\mathbf{X}, \mathbf{Y}) \right] \right\} \\
&= \mathbb{E} G_{\phi,z,\lambda,\bar{\imath}}^2 (\mathbf{X}, \mathbf{Y}) \\
&= \mathrm{Var} (G_{\phi,z,\lambda,\bar{\imath}} (\mathbf{X}, \mathbf{Y})),
\end{align*}
$$

whence

$$
\mathrm{Var} \left( \pi_k \left( G_{\phi,z,\lambda,\bar{\imath}} - G'_{\phi,z,\lambda,\bar{\imath}} \right) \circ \varphi \right) \le \mathrm{Var} \left[ \left( G_{\phi,z,\lambda,\bar{\imath}} - G'_{\phi,z,\lambda,\bar{\imath}} \right) \circ \varphi \right]. \quad (68)
$$

Thus

$$
\begin{align*}
\sigma &= \left( \sup_{G, G' \in \mathcal{G}} \operatorname{Var} \left[ \pi_k \left( G_{\phi,z,\lambda,\bar{\imath}} - G'_{\phi,z,\lambda,\bar{\imath}} \right) \circ \varphi \right] \right)^{1/2} \\
&\le \left( \sup_{G, G' \in \mathcal{G}} \operatorname{Var} \left[ \left( G_{\phi,z,\lambda,\bar{\imath}} - G'_{\phi,z,\lambda,\bar{\imath}} \right) \circ \varphi \right] \right)^{1/2} \\
&= \sup_{G, G' \in \mathcal{G}} \rho_{\mathbb{L},2}(G, G'). \tag{69}
\end{align*}
$$

Let us apply Theorem A.3 to the class

$$
\mathcal{C} = \left\{ G_{\phi,z,\lambda,\bar{\imath}} - G'_{\phi,z,\lambda,\bar{\imath}} : G_{\phi,z,\lambda,\bar{\imath}}, G'_{\phi,z,\lambda,\bar{\imath}} \in \mathcal{G} \text{ and } \rho_{\mathbb{L},2}(G_{\phi,z,\lambda,\bar{\imath}}, G'_{\phi,z,\lambda,\bar{\imath}}) \le t^{1/2} n^{-a/2} (\lambda h_n^p)^{-a/2} \right\}. \quad (70)
$$

From (69), we have

$$
\sigma \le \sup_{f,g \in \mathcal{G}} \rho_{\mathbb{L},2}(f,g) \le t^{1/2} n^{-a/2} (\lambda h_n^p)^{-a/2}. \quad (71)
$$

Since there exists a constant $\mathcal{C} > 0$ such that for any $t > \mathcal{C} \log n$

$$
t n^{-a/2} (\lambda h_n^p)^{-a/2} > (\mathfrak{C} n^{-1/2} \log n) \bigvee (\mathfrak{C} \sigma (\log 1/\sigma)^{1/2}),
$$

it follows, by selecting appropriate constants and invoking Theorem A.3, that

$$
\mathbb{P} \left( \omega \left( W_n; \rho_{\mathbb{L},2}; \frac{t^{1/2}}{n^{a/2} (\lambda h_n^p)^{a/2}} \right) \geq t n^{-a/2} (\lambda h_n^p)^{-a/2} \right)
$$
$$
\begin{align}
&= \mathbb{P} \left( \sup_{f \in \mathcal{C}} |W_n(f)| \geq tn^{-\alpha/2} (\lambda h_n^p)^{-\alpha/2} \right) \nonumber \\
&\leq d_5 \exp \left( -t^2 / (d_6 \sigma^2 n^\alpha (\lambda h_n^p)^\alpha) \right) \bigvee d_7 \exp \left( -tn^{(1-\alpha)/2} (\lambda h_n^p)^{-\alpha/2} / d_8 \right) \nonumber \\
&\leq c_1 \exp \left( -t/c_2 \right). \tag{72}
\end{align}
$$

Hence the proof is complete.

<figure><img src="image_2.png" /></figure>

Data availability

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

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