# Adaptive estimation for nonparametric circular regression with errors in variables

Tien Dat Nguyen, Thanh Mai Pham Ngoc

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Tien Dat Nguyen, Thanh Mai Pham Ngoc. Adaptive estimation for nonparametric circular regression with errors in variables. 2025. hal-05223045

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# Adaptive estimation for nonparametric circular regression with errors in variables

Tien Dat Nguyen *†, Thanh Mai Pham Ngoc‡

August 26, 2025

## Abstract

This paper investigates the nonparametric estimation of a circular regression function in an errors-in-variables framework. Two settings are studied, depending on whether the covariates are circular or linear. Adaptive estimators are constructed and their theoretical performance is assessed through convergence rates over Sobolev and Hölder smoothness classes. Numerical experiments on simulated and real datasets illustrate the practical relevance of the methodology.

**Keywords**: Circular data, Errors-in-variables model, Adaptive estimation, Deconvolution. MSC 2010. Primary 62G08, secondary 62H11.

## 1 Introduction

We consider the problem of circular regression with errors in variables. We observe the i.i.d. dataset $(Z_1, \Theta_1), \dots, (Z_n, \Theta_n)$ where

$$ \Theta_i = m(X_i) + \zeta_i \quad (\text{mod } 2\pi). $$

and

$$ Z_i = X_i + \varepsilon_i, $$

with the responses $\Theta_i$ and the regression errors $\zeta_i$ lying in $\mathbb{S}^1$, the unit circle in $\mathbb{R}^2$.

We consider two scenarios: the first where the covariates $X_i$ belong to $\mathbb{S}^1$ (circular predictor case), and the second where the $X_i$ lie in $[0, 1]$ (linear predictor case). The predictors $X_i$ are latent, and the covariate errors $\varepsilon_i$ are i.i.d. unobservable random variables with density $f_\varepsilon$, whose characteristic function is assumed to be known. The $\varepsilon_i$ are independent of $X_i$. Our objective is to estimate the circular regression function $m$ nonparametrically from the i.i.d. dataset $(Z_1, \Theta_1), \dots, (Z_n, \Theta_n)$. The term circular data is used to distinguish such data from those supported on the real line (or its subsets), henceforth referred to as linear data.

Circular or angular data are encountered in various scientific fields, such as biology (e.g., directions of animal migration), bioinformatics (e.g., protein conformational angles), geology (e.g., rock fracture orientations), medicine (e.g., circadian rhythms), forensics (e.g., crime timing), and the social sciences (e.g., time-of-day or calendar effects). Comprehensive surveys of statistical methods for circular data can be found in Mardia and Jupp [31], Jammalamadaka and SenGupta [21], Ley and Verdebout [29], and recent advances are compiled in Pewsey and García-Portugués [35].

<footnote>*Faculty of Mathematics and Computer Science, Vietnam National University Ho Chi Minh city, Vietnam; E-mail: ndat@hcmus.edu.vn</footnote>

<footnote>†Vietnam National University Ho Chi Minh city, Vietnam</footnote>

<footnote>‡LAGA, UMR 7539, Institut Galilée, Université Sorbonne Paris Nord, 93430, Villetaneuse, France; E-mail: phamngoc@math.univ-paris13.fr</footnote>
The circular response in our model introduces substantial statistical challenges compared to classical regression with linear responses. Circular regression necessitates methodologies that respect the geometry of the circle. In the parametric setting, various ad hoc approaches have been proposed (see [20], [25], [26], [18], [27], [12], [16]). To the best of our knowledge, even without measurement errors, nonparametric circular regression has been explored in only a few works, such as [11], [34] and [22]. Only one recent study [9] addresses the case involving errors in variables. Errors-in-variables models arise in many experimental contexts where data are corrupted by measurement errors. These models have been extensively studied on the real line (see [33], [6], [28], [7], [1], [3], [23], [24]), but much less so in the circular response setting. Such models are particularly demanding as they involve a deconvolution step in the estimation process.

As previously mentioned, the only work addressing nonparametric circular regression with measurement errors is [9], hence this problem is largely unexplored in statistics. In that study, the authors propose a deconvolution kernel estimator for both the circular and linear covariate cases. Their approach depends critically on the choice of key parameters: a kernel bandwidth (related to the regression estimation) and a spectral cutoff (related to deconvolution). However, the selection of these parameters is not theoretically addressed, and no convergence rates or adaptive procedures are provided. By adaptation, we mean that the estimator does not require prior knowledge of the regularity of the regression function and can automatically adjust to it—a feature essential in practice. In their numerical implementation, the authors of [9] rely on cross-validation to select the kernel bandwidth and choose arbitrary spectral cutoff values. In contrast, we propose a novel data-driven estimation procedure for the circular regression function in both settings: circular and linear predictors. We introduce adaptive rules for bandwidth selection inspired from the Goldenshluger and Lepski method (see [19]) and deconvolution density problems (see [5]) and establish convergence rates for the pointwise risk over Sobolev and Hölder classes. Our methodology is based on concentration inequalities. Finally, we assess the performance of our estimators through both simulation studies and real data applications. We show that our statistical procedure achieves satisfactory performance.

The paper is organized as follows. Section 2 details the framework and the specific features of circular regression. Sections 3 and 4 deal respectively with the circular and linear predictor cases. Numerical experiments are presented in Section 5, and all proofs are collected in Section 6.

*Notations.* It is necessary to equip the reader with some notations.

The notation $x_+$ means $\max(0, x)$. For two integers $a, b$ we denote $a \wedge b := \min(a, b)$. For two functions $u, v$ we denote $u(x) \lesssim v(x)$ if there exists a positive constant $C$ not depending on $x$ such that $u(x) \le Cv(x)$ and $u(x) \approx v(x)$ if $u(x) \lesssim v(x)$ and $v(x) \lesssim u(x)$.

In Section 3, $\| \cdot \|_{\ell_p}$, $p \ge 1$ denotes the norm on complex sequences space

$$ \|x\|_{\ell_p} := \left( \sum_{l \in \mathbb{Z}} |x_l|^p \right)^{1/p}, $$

and the $l$-th Fourier coefficient of a function $f \in L^1(S^1)$ is defined as $f^*(l) := \int_{S^1} f(x)e^{ilx}dx$, $l \in \mathbb{Z}$.

In Section 4, $\| \cdot \|_{L^1(\mathbb{R})}$ and $\| \cdot \|_{L^2(\mathbb{R})}$ respectively denote the $L^1$ and $L^2$ norm on $\mathbb{R}$ with respect to the Lebesgue measure:

$$ \|f\|_{L^1(\mathbb{R})} = \int_{\mathbb{R}} |f(y)|dy, \quad \|f\|_{L^2(\mathbb{R})} = \left( \int_{\mathbb{R}} |f(y)|^2 dy \right)^{1/2}, $$

and the $L^\infty$ norm is defined by $\|f\|_\infty = \sup_{y \in \mathbb{R}} |f(y)|$. And for $f \in L^1(\mathbb{R})$, $f^*(x) := \int e^{ixt} f(t)dt$, $x \in \mathbb{R}$ denotes the Fourier transform of $f$. Moreover, we denote $*$ the classical convolution product defined for functions $f, g$ by $f * g(x) := \int_{\mathbb{R}} f(x - y)g(y)dy$, $x \in \mathbb{R}$.
In the sequel, a point on $\mathbb{S}^1$ will not be represented as a two-dimensional vector $\mathbf{w} = (w_2, w_1)^\top$ with unit Euclidean norm but as an angle $\theta = \operatorname{atan2}(w_1, w_2)$ where

**Definition 1.1.** The function atan2: $\mathbb{R}^2 \setminus (0,0) \mapsto [-\pi; \pi]$ is defined for any $(w_1, w_2) \in \mathbb{R}^2 \setminus (0,0)$ by

$$ \operatorname{atan2}(w_1, w_2) := \begin{cases} \arctan\left(\frac{w_1}{w_2}\right) & \text{if } w_2 \ge 0, w_1 \ne 0 \\ 0 & \text{if } w_2 > 0, w_1 = 0 \\ \arctan\left(\frac{w_1}{w_2}\right) + \pi & \text{if } w_2 < 0, w_1 > 0 \\ \arctan\left(\frac{w_1}{w_2}\right) - \pi & \text{if } w_2 < 0, w_1 \le 0, \end{cases} $$

with arctan taking values in $[-\pi/2, \pi/2]$. In particular for $w_1 > 0$, $\operatorname{atan2}(w_1, 0) = \arctan(+\infty) = \pi/2$ and $\operatorname{atan2}(-w_1, 0) = \arctan(-\infty) = -\pi/2$.

In this definition, one has arbitrarily fixed the origin of $\mathbb{S}^1$ at $(1,0)^\top$ and uses the anti-clockwise direction as positive. Thus, a circular random variable can be represented as angle over $[-\pi, \pi)$. Observe that $\operatorname{atan2}(0,0)$ is not defined.

## 2 Regression with a circular response

Circular data are fundamentally different from linear data due to their periodicity, and thus require specific techniques. To measure the closeness between two angles $\theta_1$ and $\theta_2$, we do not consider the natural distance

$$ d(\theta_1, \theta_2) := \min \{|\theta_1 - \theta_2 + 2k\pi| : k \in \mathbb{Z}\}, \quad \theta_1, \theta_2 \in [-\pi, \pi), $$

but focus on $d_c$ with

$$ d_c(\theta_1, \theta_2) := 1 - \cos(\theta_1 - \theta_2), \quad \theta_1, \theta_2 \in [-\pi, \pi), $$

which is extensively used in the literature of directional statistics (see for instance Section 2 in the seminal monograph by Mardia and Jupp [31], Section 3.2.1 of [29], [11] or [32]). Note that the divergence $d_c$ corresponds to the usual squared Euclidean norm in $\mathbb{R}^2$. Indeed, the angles $\theta_1$ and $\theta_2$ determine the corresponding points $(\cos \theta_1, \sin \theta_1)$ and $(\cos \theta_2, \sin \theta_2)$ respectively on the unit circle $\mathbb{S}^1$. Then, the usual squared Euclidean norm in $\mathbb{R}^2$ reads

$$ (\cos \theta_1 - \cos \theta_2)^2 + (\sin \theta_1 - \sin \theta_2)^2 = 2 \cdot [1 - \cos(\theta_1 - \theta_2)] = 2 \cdot d_c(\theta_1, \theta_2). $$

Hence, $\sqrt{d_c}$ is a distance on $[-\pi, \pi)$ and we naturally look for a measurable function $m$ such that:

$$ \mathbb{E}[d_c(\Theta, m(X))] = \min_{f: \Omega \to [-\pi, \pi)} \mathbb{E}[d_c(\Theta, f(X))], \quad (1) $$

where $\Omega$ denotes either $[0, 1]$ or $\mathbb{S}^1$ depending on the nature of the predictors $X_i$ and the minimum is taken over $[-\pi, \pi)$-valued functions $f$ that are measurable with respect to the $\sigma$-algebra generated by $X$. It is interesting to notice that the minimization problem (1) is directly linked to the definition of the Frechet mean on the circle (see [2]). Furthermore, in directional statistics, the problem of finding such a regression function $m(X)$ as defined in (1) has been already considered (see [34] and [32]).

Now let us work conditionally to $X$. For $x \in \Omega$, let

$$ m_1(x) := \mathbb{E}(\sin(\Theta)|X = x) \quad \text{and} \quad m_2(x) := \mathbb{E}(\cos(\Theta)|X = x). \quad (2) $$

Moreover, write for an arbitrary function $f: \Omega \to [-\pi, \pi)$

$$ \begin{aligned} \mathbb{E}[\cos(\Theta - f(X))|X] &= \cos(f(X))m_2(X) + \sin(f(X))m_1(X) \\ &= \sqrt{(m_2(X))^2 + (m_1(X))^2} \cos(f(X) - \gamma(X)), \end{aligned} $$
where $\gamma: \Omega \to [-\pi, \pi)$ is defined for $x \in \Omega$

$$ \cos(\gamma(x)) := \frac{m_2(x)}{\sqrt{(m_2(x))^2 + (m_1(x))^2}} \quad \text{and} \quad \sin(\gamma(x)) := \frac{m_1(x)}{\sqrt{(m_2(x))^2 + (m_1(x))^2}}. $$

Observe that

$$ \gamma(x) = \operatorname{atan2}(m_1(x), m_2(x)). $$

Thus, we have

$$ \begin{aligned} \min_{f: \Omega \to [-\pi, \pi)} \mathbb{E}[d_c(\Theta, f(X))] &= 1 - \max_{f: \Omega \to [-\pi, \pi)} \mathbb{E}[\mathbb{E}[\cos(\Theta - f(X))|X]] \\ &= 1 - \max_{f: \Omega \to [-\pi, \pi)} \mathbb{E}[\sqrt{m_1^2(X) + m_2^2(X)} \cos(f(X) - \gamma(X))]. \end{aligned} $$

Finally the minimizer of the minimization problem (1) is achieved for

$$ f(x) = \gamma(x) = \operatorname{atan2}(m_1(x), m_2(x)). $$

In conclusion, the circular nature of the response is taken into account by the arctangent of the ratio of the conditional expectation of sine and cosine components of $\Theta$ given $X$. Accordingly, we tackle the regression problem by estimating the function

$$ m(x) = \operatorname{atan2}(m_1(x), m_2(x)), \quad x \in \Omega, \qquad (3) $$

with $m_1$ and $m_2$ defined in (2).

**Remark 2.1.** Observe that if $m_1(x) = m_2(x) = 0$, then $m(x)$ is not defined. This occurs if and only if

$$ \phi_1(f(\cdot|x)) := \int_{-\pi}^{\pi} e^{i\theta} f(\theta|x) d\theta = 0, $$

where $f(\cdot|x)$ denotes the conditional density of $\Theta|X = x$. Note that $\phi_1(f(\cdot|x))$ plays a specific role in the literature of directional statistics. See for instance Section 3.4.2 of [31].

In the sequel, we estimate the circular regression function $m$ as defined in (3) under the condition

$$ \phi_1(f(\cdot|x)) \neq 0. \qquad (4) $$

Since the function $\operatorname{atan2}(w_1, w_2)$ is undefined when $w_1 = w_2 = 0$, it is reasonable to consider the following assumption:

**Assumption 2.2.**

$$ m_1(x) \neq 0 \quad \text{or} \quad m_2(x) \neq 0, \quad \text{for } x \in \Omega, $$

and let $\delta > 0$ be defined as

$$ \delta = \begin{cases} \min(|m_1(x)|, |m_2(x)|) & \text{if } m_1(x) \neq 0 \text{ and } m_2(x) \neq 0, \\ |m_1(x)| & \text{if } m_2(x) = 0, \\ |m_2(x)| & \text{if } m_1(x) = 0. \end{cases} \qquad (5) $$

**Assumption 2.3.** Assume that there exists $\delta_X > 0$ such that

$$ f_X(x) \geq \delta_X, \quad \text{for all } x \in \Omega. \qquad (6) $$
Now let us expose the overall estimation strategy. In regression with random design, the classical idea to estimate the regression functions $m_i$ with $i \in \{1, 2\}$ is to write $m_i$ as a ratio $\frac{m_i f_X}{f_X}$, with $f_X$ the density of $X$. Hence one usually considers estimators of intermediate functions $p_i$ defined as:

$$p_i := m_i f_X, \quad i \in \{1, 2\},$$

along with an estimator of $f_X$ obtained by a deconvolution step due to the errors-in-variables setting. In constrat, here in our circular response setting, the estimator of $f_X$ does not have any impact because of the ratio involved in the atan2 function (see Definition 1.1) which entails that atan2($m_1, m_2$) = atan2($p_1, p_2$). This is actually a specificity of the circular response regression compared with the linear one. Accordingly, our strategy will consist in, first estimating $p_1$ and $p_2$ and then $m = atan2(p_1, p_2)$.

# 3 Circular predictor

## 3.1 Model

We begin by considering the case where the predictor is circular, which represents the most challenging and novel setting. More specifically, we study the following model:

$$\Theta = m(X) + \zeta \pmod{2\pi},$$

with $\Theta, X$ and $\zeta \in \mathbb{S}^1$. The $X$ are latent and we only have access to:

$$Z = X + \varepsilon \pmod{2\pi}.$$

We assume that $X$ and $\varepsilon$ are independent and the characteristic function of the density $f_\varepsilon$ is known. From an i.i.d dataset $(Z_1, \Theta_1), \dots, (Z_n, \Theta_n)$, we aim to estimate $m: \mathbb{S}^1 \to \mathbb{S}^1$.

## 3.2 Assumptions

We shall present our results in terms of Sobolev classes. Due to circular covariates, we deal with Fourier series instead of Fourier transforms. For any function $f(x) = \frac{1}{2\pi} \sum_{l \in \mathbb{Z}} f^*(l) e^{-ilx}$ in $L^2(\mathbb{S}^1)$ let

$$\|f\|_{W_\beta}^2 := \sum_{l \in \mathbb{Z}} |f^*(l)|^2 (1 + l^2)^\beta.$$

We define the class $W(\beta, S)$ to be

$$W(\beta, S) = \{f \in L^2(\mathbb{S}^1) : \|f\|_{W_\beta}^2 \le S^2\}. \qquad (7)$$

Following [8], we make the following assumptions about the noise in the variables.

**Assumption 3.1.** The noise $\varepsilon$ is called ordinary smooth (OS) if there exist $\nu, a_0, a_1 > 0$, such that

$$a_0(1 + |l|)^{-\nu} \le |f_{\varepsilon}^*(l)| \le a_1(1 + |l|)^{-\nu}, \quad l \in \mathbb{Z}. \qquad (8)$$

The noise $\varepsilon$ is called supersmooth (SS) if there exist $a_0, a_1, b, a > 0$ and $c \in \mathbb{R}$

$$a_0(|l| + 1)^c \exp(-b|l|^a) \le |f_{\varepsilon}^*(l)| \le a_1(|l| + 1)^c \exp(-b|l|^a), \quad l \in \mathbb{Z}. \qquad (9)$$

Let us recall that some well-known circular densities belong to these two types of noise. Examples of supersmooth densities include the densities of wrapped Normal and wrapped Cauchy distributions; conversely, the wrapped Laplace and the wrapped Gamma densities are examples of ordinary smooth ones.
## 3.3 Preliminary estimators

As explained at the end of Section 2, the target function $m$ is given by $m = \operatorname{atan2}(m_1, m_2) = \operatorname{atan2}(p_1, p_2)$ with $p_1(x) = \mathbb{E}(\sin \Theta_1 | X = x) f_X(x)$ and $p_2(x) = \mathbb{E}(\cos \Theta_1 | X = x) f_X(x)$. We first aim to find estimates of $p_1$ and $p_2$. To this end, we adopt an orthogonal projection method onto the Fourier basis of $\mathbb{L}^2(\mathbb{S}^1)$.

In the nonparametric framework, the usual idea is to find an estimator of the $L$-th projection of $p_i, i \in \{1, 2\}$ namely an estimator of

$$p_{i,L_i}(x) := \frac{1}{2\pi} \sum_{|l| \le L_i} p_i^*(l) e^{-ilx}, x \in \mathbb{S}^1.$$

We propose the following estimates for $p_1$ and $p_2$:

$$\hat{p}_{1,L_1}(x) = \sum_{|l| \le L_1} \frac{1}{2\pi n} \sum_{j=1}^{n} \sin(\Theta_j) \frac{e^{ilZ_j}}{f_{\varepsilon}^*(l)} e^{-ilx}, x \in \mathbb{S}^1,$$

$$\hat{p}_{2,L_2}(x) = \sum_{|l| \le L_2} \frac{1}{2\pi n} \sum_{j=1}^{n} \cos(\Theta_j) \frac{e^{ilZ_j}}{f_{\varepsilon}^*(l)} e^{-ilx}, x \in \mathbb{S}^1.$$

The expression of $\hat{p}_{1,L_1}(x)$ (same line of reasoning for $\hat{p}_{2,L_2}(x)$) is justified by the fact that

$$\mathbb{E}[\hat{p}_{1,L_1}(x)] = \frac{1}{2\pi} \sum_{|l| \le L_1} \frac{e^{-ilx}}{f_{\varepsilon}^*(l)} \mathbb{E}(\mathbb{E}(\sin(\Theta_1)e^{il(X_1+\varepsilon_1)}|X_1))) = \frac{1}{2\pi} \sum_{|l| \le L_1} e^{-ilx} \mathbb{E}(e^{ilX_1}m_1(X_1)) \\ = \frac{1}{2\pi} \sum_{|l| \le L_1} p_1^*(l) e^{-ilx} = p_{1,L_1}(x),$$

which entails that $\hat{p}_{1,L_1}(x)$ is a unbiased estimator of $p_{1,L_1}(x)$.

### 3.3.1 Bias and Variance

We have the following bias-variance decomposition of the pointwise risk of $\hat{p}_{i,L_i}(x)$:

$$\mathbb{E}|\hat{p}_{i,L_i}(x) - p_i(x)|^2 = \underbrace{\mathbb{E}|\hat{p}_{i,L_i}(x) - \mathbb{E}[\hat{p}_{i,L_i}(x)]|^2}_{=: \operatorname{Var}(\hat{p}_{i,L_i}(x))} + |\mathbb{E}[\hat{p}_{i,L_i}(x)] - p_i(x)|^2.$$

**Proposition 3.2.** For $i \in \{1, 2\}$, if $p_i \in W(\beta_i, S_i)$ with $\beta_i > \frac{1}{2}$, the bias is controlled as follows

$$|\mathbb{E}[\hat{p}_{i,L_i}(x)] - p_i(x)| = \frac{1}{2\pi} \left| \sum_{|l| \le L_i} p_i^*(l) \right| \le C(S_i, \beta_i) L_i^{-\beta_i},$$

with $C(S_i, \beta_i)$ a constant depending on $S_i$ and $\beta_i$. The variance is controlled by

$$\operatorname{Var}(\hat{p}_{i,L_i}(x)) \le V_0(n, L_i)$$

with

$$V_0(n, L_i) := \frac{1}{(2\pi)^2 n} \min \left\{ \left( \sum_{|l| \le L_i} \frac{1}{|f_{\varepsilon}^*(l)|} \right)^2, \|f_{\varepsilon}^*\|_{\ell_1} \sum_{|l| \le 2L_i} \frac{1}{|f_{\varepsilon}^*(l)|^2} \right\}.$$

Furthermore, we have that
1. If the noise $\varepsilon$ is OS (satisfying (8)) with $\nu > 1$ and provided that $L_i \ge 1$, for $i \in \{1, 2\}$, we have

$$ \mathrm{Var}(\hat{p}_{i,L_i}(x)) \le \frac{\max\{1, \|f_{\varepsilon}^{*}\|_{\ell_1}\}}{(2\pi)^2 n} L_i^{2\nu+1}. $$

2. If the noise $\varepsilon$ is SS (satisfying (9)), we have

$$ \mathrm{Var}(\hat{p}_{i,L_i}(x)) \le \frac{\max\{1, \|f_{\varepsilon}^{*}\|_{\ell_1}\}}{(2\pi)^2 n} L_i^{-2c+1} e^{2bL_i^a}. $$

The proof of Proposition 3.2 is given in Section 6.1.1.

### 3.3.2 Rates of convergence

When the noise $\varepsilon$ is ordinary smooth and the functions $p_i$ belong to a Sobolev class $W(\beta_i, S_i)$, $i \in \{1, 2\}$, we deduce from Proposition 3.2, via the usual bias-variance tradeoff, that the optimal choice of the projection level is given by:

$$ L_{i,\mathrm{opt},\mathrm{OS}} = \arg \min_{L \in \mathbb{N}^*} \left\{ L^{-2\beta_i} + \frac{1}{n} L^{2\nu+1} \right\} \Rightarrow L_{i,\mathrm{opt},\mathrm{OS}} \propto n^{\frac{1}{2\beta_i+2\nu+1}}, $$

and the rate of convergence for the pointwise risk to estimate $p_i$ is $n^{-\frac{2\beta_i}{2\beta_i+2\nu+1}}$. This corresponds to the usual minimax rate of convergence for estimating a univariate function of regularity $\beta_i$ in an ordinary ill-posed inverse problem of severity $\nu$ (see Chapter 3 in [33]). One remarks that the optimal level $L_{i,\mathrm{opt},\mathrm{OS}}$ depends on the regularity $\beta_i$ which is unknown. This problem will be tackle in Section 3.4 where we propose a data-driven procedure to select $L_i$.

When the noise $\varepsilon$ is supersmooth, and the functions $p_i$ belong to a Sobolev class $W(\beta_i, S_i)$, again from Proposition 3.2, we deduce that the optimal levels are given by:

$$ L_{i,\mathrm{opt},\mathrm{SS}} = \arg \min_{L \in \mathbb{N}^*} \left\{ L^{-2\beta_i} + \frac{1}{n} L^{-2c+1} e^{2bL^a} \right\} \Rightarrow L_{i,\mathrm{opt},\mathrm{SS}} \propto L_{i,\mathrm{opt},\mathrm{SS}} := \left( \frac{\log n}{2b} \right)^{\frac{1}{a}}. \quad (10) $$

The subsequent rates of convergence for the pointwise risk to estimate $p_i$ are $(\log n)^{-\frac{2\beta_i}{a}}$.

In the case of a supersmooth noise, unlike in the OS noise setting, the optimal level $L_{i,\mathrm{opt},\mathrm{SS}}$ does not depend on the regularity of the functions $p_i$, but solely on the noise parameters, which are assumed to be known. As a consequence, no level selection procedure is required in this case.

## 3.4 Adaptive estimation

As shown in the previous section, when the noise is ordinary smooth, it is necessary to design a data-driven procedure for selecting the levels $L_1$ and $L_2$. To this end, we employ a Goldenshluger-Lepski selection rule adapted to projection methods and indirect observations (see [4] and [5]). Consider the following collection of levels:

$$ \mathcal{L} = \left\{ L \in \{1, \dots, n\}, \frac{\sum_{|l| \le 2L} \frac{1}{|f_{\varepsilon}^{*}(l)|^2}}{\left(\sum_{|l| \le L} \frac{1}{|f_{\varepsilon}^{*}(l)|}\right)^2} \ge \frac{\log n}{n} \right\}. \quad (11) $$

Note that condition expressed in (11) amounts to have $L \lesssim \frac{n}{\log n}$.
For $i \in \{1, 2\}$, let us set

$$A(L_i, x) = \sup_{L'_i \in \mathcal{L}} \left\{ |\hat{p}_{i,L'_i}(x) - \hat{p}_{i,L_i \wedge L'_i}(x)| - \sqrt{V(n, L'_i)} \right\}_+, \quad (12)$$

with

$$V(n, L_i) = c_{0,i} \log(n) V_0(n, L_i),$$

and $c_{0,i} > 0$ is a tuning constant which will be specified later while $V_0(n, L_i)$ has been defined in Proposition 3.2.

Finally, we define our selected levels as

$$\hat{L}_i := \arg \min_{L_i \in \mathcal{L}} \left\{ A(L_i, x) + \sqrt{V(n, L_i)} \right\}. \quad (13)$$

We now obtain the following theorem which states an oracle inequality.

**Theorem 3.3.** Suppose that the noise $\varepsilon$ is OS (satisfying (8)). Let $q \ge 1$ and $\min\{c_{0,1}, c_{0,2}\} \ge \frac{64(2+q)^2}{\min(1, \|f_{\varepsilon}^*\|_{\ell_1})}$. Then with probability larger than $1 - 4n^{-q}$ we have

$$|\hat{p}_{i, \hat{L}_i}(x) - p_i(x)| \le \inf_{L_i \in \mathcal{L}} \left\{ \frac{1}{\pi} \sum_{|l| \ge L_i} |p_i^*(l)| + \frac{5}{2} \sqrt{V(n, L_i)} \right\}, \quad i \in \{1, 2\}. \quad (14)$$

The proof of Theorem 3.3 is given in Section 6.1.2.

Theorem 3.3 highlights the bias-variance decomposition of the pointwise risk. The first term of the r.h.s is a bias term (see Proposition 3.2) while the second term is a variance term.

### 3.4.1 Rates of convergence for $m$

In this section, we derive rates of convergence for estimating the target regression function $m$.

In the case of an ordinary smooth noise, we set the final estimator of $m$ to be:

$$\hat{m}_{\hat{L}}(x) := \operatorname{atan2}(\hat{p}_{1, \hat{L}_1}(x), \hat{p}_{2, \hat{L}_2}(x)). \quad (15)$$

In the case of a supersmooth noise, remind that no selection level is required (see Section 3.3.2) and we have adaptation for free. Then, the estimator of $m$ is defined as:

$$\hat{m}_{L_{opt,SS}}(x) := \operatorname{atan2}(\hat{p}_{1,L_{opt,SS}}(x), \hat{p}_{2,L_{opt,SS}}(x)),$$

with $L_{opt,SS}$ defined in (10).

The oracle inequality stated in Theorem 3.3 serves as a key tool for deriving convergence rates for the estimation of $m$ in the presence of ordinary smooth noise. In contrast, obtaining convergence rates in the supersmooth noise setting is a straightforward consequence of the proofs established for the ordinary smooth case. The next theorem presents the convergence rates.

**Theorem 3.4.** Let $p_1$ belongs to $W(\beta_1, S_1)$ and $p_2$ belongs to $W(\beta_2, S_2)$.

1. Assume that the noise $\varepsilon$ is OS (satisfying (8)) with $\nu > 1$. Let $q \ge 1$ and suppose that $\min\{c_{0,1}; c_{0,2}\} \ge \frac{64(2+q)^2}{\min(1, \|f_{\varepsilon}^*\|_{\ell_1})}$. Then, for $n$ sufficiently large,

$$\mathbb{E}[d_c(\hat{m}_{\hat{L}}(x), m(x))] \le \frac{C_1}{\delta^2} \max\{\psi_n^2(\beta_1, \nu), \psi_n^2(\beta_2, \nu)\},$$

where the convergence rates are $\psi_n(\beta_i, \nu) = (\log n/n)^{\frac{\beta_j}{2\beta_i+2\nu+1}}$, the constant $\delta$ is defined in (5) and $C_1$ is a constant depending on $\beta_1, \beta_2, S_1, S_2, c_{0,1}, c_{0,2}$ and $f_{\varepsilon}^*$.
2. Assume that the noise $\epsilon$ is SS (satisfying (9)). For $n$ sufficiently large,

$$ \mathbb{E}[d_c(\hat{m}_{L_{opt,SS}}(x), m(x))] \le \frac{C_2}{\delta^2} \max\{\tilde{\psi}_n^2(\beta_1, a), \tilde{\psi}_n^2(\beta_2, a)\}, $$

with $\tilde{\psi}_n(\beta_i, a) = (\log n)^{-\frac{\beta_i}{a}}$, the constant $\delta$ defined in (5) and $C_2$ is a constant depending on $\beta_1, \beta_2, S_1, S_2$ and $f_{\epsilon}^*.$

The proof of Theorem 3.4 is given in Section 6.1.3.

**Remark 3.5.** Note that when $\beta_1 = \beta_2 = \beta$, the resulting rates are $(\log n/n)^{\frac{2\beta}{2\beta+2\nu+1}}$ and $(\log n)^{-2\beta/a}$, which correspond to the optimal rates for adaptive estimation of a univariate regression function and for pointwise risk in an errors-in-variables framework (see Section 3.3.2 in [33]). The presence of a logarithmic factor in the rate under ordinary smooth noise is expected, since we deal with pointwise adaptive estimation.

# 4 Linear predictor

In this section, we consider the case where the predictor $X \in [0, 1]$.

## 4.1 Model

We study the following model:

$$ \Theta = m(X) + \zeta \pmod{2\pi} $$

with $\Theta, \zeta \in \mathbb{S}^1$ and $X \in [0, 1]$. The variable $X$ is unobserved and we only have access to

$$ Z = X + \epsilon. $$

We assume that $X$ and $\epsilon$ are independent and the characteristic function of density the $f_{\epsilon}$ is known. From an i.i.d dataset $(Z_1, \Theta_1), \dots, (Z_n, \Theta_n)$, we want to estimate $m: [0, 1] \mapsto \mathbb{S}^1$.

### 4.1.1 Assumptions

We present our results in terms of Hölder classes that are adapted to local estimation.

**Definition 4.1.** Let $\beta > 0$ and $\Lambda > 0$. The Hölder class $\mathcal{H}(\beta, \Lambda)$ is the set of functions $f: (0, 1) \to \mathbb{R}$, such that $f$ admits derivatives up to the order $\lfloor \beta \rfloor$, and for any $(y, \tilde{y}) \in (0, 1)^2$,

$$ \left| \frac{d^{\lfloor \beta \rfloor} f}{(dy)^{\lfloor \beta \rfloor}}(\tilde{y}) - \frac{d^{\lfloor \beta \rfloor} f}{(dy)^{\lfloor \beta \rfloor}}(y) \right| \le \Lambda |\tilde{y} - y|^{\beta - \lfloor \beta \rfloor}. $$

We make the following assumptions on the covariates’ noise.

**Assumption 4.2.** The noise $\epsilon$ is called ordinary smooth (OS) if there exist $r, c_{\epsilon}, C_{\epsilon} > 0$ such that

$$ c_{\epsilon}(1 + |t|)^{-r} \le |f_{\epsilon}^{*}(t)| \le C_{\epsilon}(1 + |t|)^{-r}, \quad t \in \mathbb{R}. \qquad (16) $$

The noise $\epsilon$ is called supersmooth (SS) if there exist constants $c_{\epsilon} > 0$, $C_{\epsilon} > 0$, $\gamma > 0$, $\rho > 0$ and $\rho_0, \rho_1 \in \mathbb{R}$ such that

$$ c_{\epsilon}(1 + |t|^2)^{-\rho_0/2} \exp(-\gamma |t|^{\rho_1}) \le |f_{\epsilon}^{*}(t)| \le C_{\epsilon}(1 + |t|^2)^{-\rho_1/2} \exp(-\gamma |t|^{\rho_0}), \quad t \in \mathbb{R}. \qquad (17) $$
Due to the linear covariates, we shall use kernel estimation.

**Definition 4.3.** Let $K: \mathbb{R} \to \mathbb{R}$ be an integrable function on $\mathbb{R}$. We say that $K$ is a kernel if it satisfies $\int_{\mathbb{R}} K(y)dy = 1$. For $h > 0$, we define $K_h(\cdot) := \frac{1}{h}K(\frac{\cdot}{h})$. Also, define

$$ M(K) := \begin{cases} \|K\|_1, & \text{if } \|K\|_1 < \infty \\ \|K^*\|_{\infty}, & \text{otherwise} \end{cases}. \qquad (18) $$

We now consider the following assumptions on the kernel $K$, following [5]. A classical one to control the bias is:

**Assumption 4.4.** The kernel $K$ is of order $\kappa \in \mathbb{R}_+$, i.e.

$$ (i) \quad C_{K,\kappa} := \int_{\mathbb{R}} (1 + |y|)^{\kappa} |K(y)| dy < \infty; $$

$$ (ii) \quad \forall k \in \{1, \dots, \lfloor \kappa \rfloor\}, \quad \int_{\mathbb{R}} y^k K(y) dy = 0. $$

Another one to ensure the estimators remain finite is:

**Assumption 4.5.** If $f_{\varepsilon}^{\star}$ satisfies Assumption (16), we assume there exists $C_{(K^{\star})} > 0$ (for simplicity, we omit the dependence on $r$ in the notation of $C_{(K^{\star})}$) such that

$$ \int_{\mathbb{R}} |K^*(t)|(1+|t|)^r dt < \sqrt{C_{(K^*)}} \quad \text{and} \quad \int_{\mathbb{R}} |K^*(t)|^2 (1+|t|)^{2r} dt < C_{(K^*)}. $$

On the other hand, if $f_{\varepsilon}^{\star}$ satisfies Assumption (17) we will consider the sinc kernel $K(y) = \frac{\sin(y)}{\pi y}$ for $y \in \mathbb{R} \setminus \{0\}$ and $K(0) = \frac{1}{\pi}$, whose Fourier Transform has a compact support, more precisely, $K^*(t) = \mathbf{1}_{[-1,1]}(t)$ for $t \in \mathbb{R}$. The sinc kernel has $M(K) = \|K^*\|_{\infty} = 1$ and satisfies Assumption 4.5.

## 4.2 Preliminary estimators

Now, with $j \in \{1; 2\}$, for $x \in [0, 1]$ and a bandwidth $h_j > 0$, we propose an estimator for $p_j$ as

$$ \hat{p}_{1,h_1}(x) := \frac{1}{n} \sum_{k=1}^{n} \left( \sin(\Theta_k) \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} e^{itZ_k} \frac{K_{h_j}^{\star}(t)}{f_{\varepsilon}^{\star}(t)} dt \right), \qquad (19) $$

$$ \hat{p}_{2,h_2}(x) := \frac{1}{n} \sum_{k=1}^{n} \left( \cos(\Theta_k) \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} e^{itZ_k} \frac{K_{h_j}^{\star}(t)}{f_{\varepsilon}^{\star}(t)} dt \right). \qquad (20) $$

The form of our estimators for $p_1$ and $p_2$ are justified by the fact that, for any $x \in [0, 1]$ and $h_1, h_2 > 0$,

$$ \mathbb{E}(\hat{p}_{1,h_1}(x)) = K_{h_1} * p_1(x) \quad \text{and} \quad \mathbb{E}(\hat{p}_{2,h_2}(x)) = K_{h_2} * p_2(x). \qquad (21) $$

Indeed we have for $x \in [0, 1]$ and $h_1 > 0$, using the independence between $X$ and $\varepsilon$ and Parseval's equality:

$$ \begin{aligned} \mathbb{E}(\hat{p}_{1,h_1}(x)) &= \mathbb{E}\left(\sin(\Theta_1) \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} e^{itZ_1} \frac{K_{h_1}^{\star}(t)}{f_{\varepsilon}^{\star}(t)} dt\right) \\ &= \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} \mathbb{E}\left(\mathbb{E}[\sin(\Theta_1) | X_1] e^{itX_1}\right) \mathbb{E}(e^{it\varepsilon_1}) \frac{K_{h_1}^{\star}(t)}{f_{\varepsilon}^{\star}(t)} dt \\ &= K_{h_1} * p_1(x), \end{aligned} $$

and the same computations handle for $\hat{p}_{2,h_2}$.
## 4.3 Bias and variance

Let us derive the bias and variance of our estimates.

**Proposition 4.6.** Let $j \in \{1, 2\}$. Suppose that $p_j$ belongs to $\mathcal{H}(\beta_j, \Lambda_j)$, with $\Lambda_j, \beta_j \in \mathbb{R}_+^*$. Suppose that the kernel $K$ satisfies Assumption 4.5 and Assumption 4.4 with an index $\kappa \in \mathbb{R}_+$ such that $\kappa \ge \beta_j$. Then

$$ \left| \mathbb{E}(\hat{p}_{j,h_j}(x)) - p_j(x) \right| \le C_{K,\kappa} \Lambda_j h_j^{\beta_j}, \quad \text{and} \quad \mathrm{Var}(\hat{p}_{j,h_j}(x)) \le \tilde{V}_0(n, h_j), $$

with the constant $C_{K,\kappa}$ defined in Assumption 4.4 and

$$ \tilde{V}_0(n, h_j) = \frac{1}{(2\pi)^2 n} \min \left\{ \left\| \frac{K_{h_j}^*}{f_\varepsilon^*} \right\|_2^2 \|f_\varepsilon^*\|_1 ; \left\| \frac{K_{h_j}^*}{f_\varepsilon^*} \right\|_1^2 \right\}. $$

Furthermore, we have that

1. If the noise $\varepsilon$ is OS (satisfying (16)) with $r > 1$, then if $h_j \le 1$,

$$ \tilde{V}_0(n, h_j) \le \frac{C}{(2\pi)^2} n^{-1} h_j^{-1-2r}. $$

2. If the noise $\varepsilon$ is SS (satisfying (17)) then

$$ \tilde{V}_0(n, h_j) \le \frac{C}{(2\pi)^2} n^{-1} h_j^{(\rho-1)+} h_j^{-2\rho_0-1+\rho} \exp \left( 2\gamma h_j^{-\rho} \right). $$

The proof of Proposition 4.6 is done in Section 6.2.1.

From Proposition 4.6, we deduce by the bias variance tradeoff the following rates of convergence to estimate the function $p_i$, which are similar to the circular covariates case:

1. If the noise $\varepsilon$ is OS, the optimal bandwidth is of order $n^{-\frac{1}{2\beta_j+2r+1}}$ which yields a convergence rate of order $n^{2\beta_j+2r+1}$. Consequently, one needs a data-driven bandwidth selection since the optimal bandwidth depends on $\beta_j$ which is unknown. This will be tackled in the next section.

2. If the noise $\varepsilon$ is SS, then the optimal bandwidth does not depend on the regularity $\beta_j$ and is of order

$$ h_{opt,SS} := \left( \frac{\log n}{2\gamma} \right)^{-\frac{1}{\rho}} \qquad (22) $$

and there is no need to select a bandwidth. The subsequent rates of convergence are of order $(\log n)^{-\frac{2\beta_j}{\rho}}$.

## 4.4 Adaptive estimation

We now focus on the case of ordinary smooth noise. Let us describe how to select in a data-driven way the bandwidths $h_1$ and $h_2$ for the estimators $\hat{p}_{1,h_1}(x)$ and $\hat{p}_{2,h_2}(x)$, respectively. This selection will be done among a convenient grid $\mathcal{H}$, defined as follows

$$ \mathcal{H} = \left\{ h = k^{-1}, k \in \mathbb{N}^*, k \le n : \left\| \frac{K_h^*}{f_\varepsilon^*} \right\|_2^2 \left( \left\| \frac{K_h^*}{f_\varepsilon^*} \right\|_1 \right)^{-2} \ge \frac{\log n}{n} \right\}. \qquad (23) $$
Again, we propose a Goldenshluger-Lepski selection procedure to select $h_1$ and $h_2$. More precisely, for $x \in [0, 1]$, we define:

$$ \tilde{A}(h_j, x) = \sup_{h'_j \in \mathcal{H}} \left\{ |\hat{p}_{j,h_j,h'_j}(x) - \hat{p}_{j,h'_j}(x)| - \sqrt{\tilde{V}_j(n, h_j)} \right\}_+ , $$

where $\tilde{V}_j(n, h_j) := \tilde{c}_{0,j} \log(n) \tilde{V}_0(n, h_j)$, $\tilde{c}_{0,j} > 0$ are tuning parameters and

$$ \hat{p}_{j,h_j,h'_j}(x) := (K_{h'_j} * \hat{p}_{j,h_j})(x) $$

so that $\hat{p}_{j,h_j,h'_j}(x) = \hat{p}_{j,h'_j,h_j}(x)$ for any $h_j, h'_j \in \mathcal{H}$. Thus, an adaptive bandwidth is selected as

$$ \hat{h}_j := \underset{h_j \in \mathcal{H}}{\operatorname{argmin}} \left\{ \tilde{A}(h_j, x) + \sqrt{\tilde{V}(n, h_j)} \right\}. \quad (24) $$

The following theorem establishes an oracle-type inequality for the estimators $\hat{p}_{j,\hat{h}_j}$.

**Theorem 4.7.** Consider the collection of bandwidths $\mathcal{H}$ defined in (23). Let $j \in \{1, 2\}$, $q \ge 1$ and assume that $\tilde{c}_{0,j} \ge 16(2+q)^2(1+M(K))^2 \frac{1}{\min\{\|f_{\varepsilon}^*\|_1, 1\}}$ with $M(K) < \infty$. Then, with probability larger than $1-4n^{-q}$,

$$ |\hat{p}_{j,\hat{h}_j}(x) - p_j(x)| \le \inf_{h_j \in \mathcal{H}} \left\{ (1+2M(K)) \|p_j - K_{h_j} * p_j\|_{\infty} + 3 \sqrt{\tilde{V}_j(n, h_j)} \right\}. $$

The proof of Theorem 4.7 is given in Section 6.2.2.

### 4.4.1 Rates of convergence for $m$

In the case where the noise $\varepsilon$ is OS, the final estimator of the target regression $m$ is

$$ \hat{m}_{\hat{h}}(x) = \operatorname{atan2}(\hat{p}_{1,\hat{h}_1}(x), \hat{p}_{2,\hat{h}_2}(x)). \quad (25) $$

In the case of a SS noise, remind that no selection level is required, and we set the bandwidths to the value $h_{opt,SS}$ defined in (22). We then define the estimator of $m$ as

$$ \hat{m}_{h_{opt,SS}}(x) := \operatorname{atan2}(\hat{p}_{1,h_{opt,SS}}(x), \hat{p}_{2,h_{opt,SS}}(x)). $$

**Theorem 4.8.** Let $\beta_1, \beta_2, \Lambda_1, \Lambda_2 > 0$. Suppose that $p_1$ belongs to $\mathcal{H}(\beta_1, \Lambda_1)$, $p_2$ belongs to $\mathcal{H}(\beta_2, \Lambda_2)$ and the kernel $K$ satisfies Assumption 4.5 and Assumption 4.4 with an index $\kappa \ge \max\{\beta_1, \beta_2\}$.

1. Assume that the noise $\varepsilon$ is OS with $r > 1$. Let $q \ge 1$ and suppose that $\min\{\tilde{c}_{0,1}; \tilde{c}_{0,2}\} \ge 16(2+q)^2(1+M(K))^2 \frac{1}{\min\{\|f_{\varepsilon}^*\|_1; 1\}}$. Then, for $n$ sufficiently large,

$$ \mathbb{E}[d_c(\hat{m}_{\hat{h}}(x), m(x))] \le \frac{\tilde{C}_1}{\delta^2} \max\{\psi_n^2(\beta_1, r); \psi_n^2(\beta_2, r)\}, $$

where the convergence rates are $\psi_n(\beta_j, r) = (\log n/n)^{\frac{\beta_j}{2\beta_j+2r+1}}$, and the constant $\delta$ is defined in (5) and $\tilde{C}_1$ is a constant depending on $\beta_1, \beta_2, \Lambda_1, \Lambda_2, c_{0,1}, c_{0,2}, K$ and $f_{\varepsilon}$.
2. Assume that the noise $\varepsilon$ is SS. Then for $n$ sufficiently large we have

$$ \mathbb{E}[d_c(\hat{m}_{h_{opt,SS}}(x), m(x))] \leq \frac{\tilde{C}_2}{\delta^2} \max\{\tilde{\psi}_n^2(\beta_1, \rho); \tilde{\psi}_n^2(\beta_2, \rho)\}, $$

with $\tilde{\psi}_n(\beta_j, \rho) = (\log n)^{-\frac{\beta_j}{\rho}}$ and the constant $\delta$ is defined in (5) and $\tilde{C}_2$ is a constant depending on $\beta_1, \beta_2, \Lambda_1, \Lambda_2, c_{0,1}, c_{0,2}, K$ and $f_\varepsilon$.

The proof of Theorem 4.8 is done in Section 6.2.3.

We obtain similar convergence rates as in the case of circular predictors. When $\beta_1 = \beta_2 = \beta$, the resulting convergence rates are $(\log n/n)^{\frac{2\beta}{2\beta+2r+1}}$ and $(\log n)^{-2\beta/\rho}$, which match the minimax optimal rates for adaptive estimation of a univariate regression function and for pointwise risk in an errors-in-variables context.

# 5 Numerical results

In this section, we illustrate numerically our procedure. Both simulations and a real data example are considered.

## 5.1 Simulations

### 5.1.1 Linear predictor case

We consider the following linear-circular regression model:

$$ \begin{array}{l} \text{(LC).} \quad \Theta = \text{atan2}(20X - 11, (10X - 5)^2 + 2) + \zeta \pmod{2\pi}, \\ Z = X + \varepsilon, \end{array} \qquad (26) $$

where the covariate $X \sim U([0, 1])$ and the regression circular error $\zeta$ is distributed according to a von Mises distribution $f_{vM(0,10)}$. Remind that the classical von Mises distribution with location parameter $\mu \in [-\pi, \pi)$ and concentration parameter $\kappa > 0$ denoted whose density $f_{vM(\mu,\kappa)}$ is defined for any $\theta \in [-\pi, \pi)$ by

$$ f_{vM(\mu,\kappa)}(\theta) = c(\kappa) \exp(\kappa \cos(\theta - \mu)), \qquad (27) $$

with $c(\kappa)$ a normalizing constant.

For the covariates' noise $\varepsilon$, we focus on the centered Laplace density with scale parameter $\sigma_\varepsilon > 0$, which has the following expression $f_\varepsilon(x) = \frac{1}{2\sigma_\varepsilon}e^{-\frac{|x|}{\sigma_\varepsilon}}$, with $x \in \mathbb{R}$. Actually, the choice of the centered Laplace density is motivated by the fact that the Fourier transform of $f_\varepsilon$ is given by $f_\varepsilon^*(t) = \frac{1}{1 + \sigma_\varepsilon^2 t^2}$, with $t \in \mathbb{R}$, which gives an example of ordinary smooth noises with degree of ill-posedness $r = 2$ (in Assumption (16)).

Figure 1 displays an illustration of our statistical setting.

As we consider an ordinary smooth noise in the covariates, we implement the estimator $\hat{m}_{\hat{h}}(x) = \text{atan2}(\hat{p}_{1,\hat{h}_1}(x), \hat{p}_{2,\hat{h}_2}(x))$ (defined in (25)), where the bandwidths $\hat{h}_1$ and $\hat{h}_2$ are selected according to the Goldenshluger-Lepski procedure (24). We consider the sinc kernel $K(x) = \frac{\sin(x)}{\pi x}$ for $x \in \mathbb{R} \setminus \{0\}$ and $K(0) = \frac{1}{\pi}$, for which the bandwidths are selected among the following collection $\mathcal{H}_n$ defined as

$$ \mathcal{H}_n := \left\{ k^{-1} : k \in \mathbb{N}, 1 \le k \le \frac{n}{\log n} \right\}. $$
<figure><img src="image_3.png" /><figcaption>Figure 1: Illustration of model (LC) with $n = 200$. Simulated data $(\Theta_i)_{i=1}^n$ are displayed in green points. The red curve represents the regression function $m$, while the blue vertical line displays the point $x = 0.2$ where we estimate $m(x)$.</figcaption></figure>

In the selection procedure, we need to tune the two parameters $\tilde{c}_{0,1}$ and $\tilde{c}_{0,2}$. To do this, we implement preliminary simulations to calibrate $\tilde{c}_{0,1}$ and $\tilde{c}_{0,2}$ by considering model (LC), and for simplicity, we consider the case $\tilde{c}_{0,1} = \tilde{c}_{0,2} = \tilde{c}_0$. We compute the following risk $\mathcal{R}$:

$$ \mathcal{R} := 1 - \cos(\hat{m}_{\hat{h}}(x), m(x)), \qquad (28) $$

and consider $\mathcal{R}$ as a function of $\tilde{c}_0$ on the following discretization grid:

$$ G_{\tilde{c}_0} := \{0.001; 0.0025; 0.005; 0.0075; 0.01; 0.025; 0.05; 0.075; 0.1; 0.2; 0.3; 0.4; 0.6; 0.8; 1; 2; 4\}. $$

The function $\mathcal{R}(\tilde{c}_0)$ is plotted in Figure 2 when estimating at $x = 0.2$, a centered Laplace distributed covariate noise Laplace(0, 0.075), a sample size $n = 200$ and 50 Monte Carlo repetitions. The figure shows a plateau phenomenon and regarding this, the choice $\tilde{c}_{0,1} = \tilde{c}_{0,2} = 0.4$ is reasonable. In the sequel, we fix $\tilde{c}_{0,1} = \tilde{c}_{0,2} = 0.4$ for subsequent numerical simulations.

<figure><img src="image_4.png" /><figcaption>Figure 2: Plot of the function $\tilde{c}_0 \in G_{\tilde{c}_0} \mapsto \mathcal{R}(\tilde{c}_0)$. The red vertical line displays the point $\tilde{c}_0 = 0.4$.</figcaption></figure>

We now illustrate the numerical performance of our procedure. Table 1 presents the ME (Mean Error) of $\hat{m}_{\hat{h}}(x)$ over 50 Monte Carlo runs with respect to the circular distance $d_c$ for model (LC)
with errors-in-variables $\varepsilon \sim \text{Laplace}(0, 0.075)$ and $\varepsilon \sim \text{Laplace}(0, 0.1)$, respectively. In addition, when dealing with regression problems with errors in the design, it is common to consider the so-called reliability ratio (see [15]) that is defined by

$$ \Upsilon = \frac{\text{Var}(X)}{\text{Var}(X) + \text{Var}(\varepsilon)}. \qquad (29) $$

This ratio $\Upsilon$ can be used to assess the amount of noise in the covariates. More specifically, when $\Upsilon$ is close to 0, the amount of noise in the covariates becomes larger and this makes the deconvolution step more difficult. We calculate the reliability ratio $\Upsilon$ corresponding to our simulations. The results in Table 1 shows that the performances of our adaptive estimator is satisfying even if the noise increases.

<table><caption>Table 1: Mean errors of $\hat{m}_{\hat{h}}(x)$ at point $x_0 = 0.2$.</caption><thead><tr><th>σ<sub>ε</sub></th><th>Υ</th><th>n = 200</th><th>n = 500</th></tr></thead><tbody><tr><td>0.075</td><td>0.88</td><td>0.0064</td><td>0.0029</td></tr><tr><td>0.10</td><td>0.80</td><td>0.0137</td><td>0.0048</td></tr></tbody></table>

### 5.1.2 Circular predictor cases

For the case of circular predictors, we consider the circular-circular regression model:

$$ \begin{array}{l} \text{(CC). } \Theta = \left( \frac{1}{2} + \cos(x + 3 \sin(X)) \right) + \zeta, \quad (\text{mod } 2\pi), \\ Z = X + \varepsilon, \quad (\text{mod } 2\pi). \end{array} $$

We consider $X \sim f_{vM(0,0.01)}$ and a regression noise $\zeta \sim f_{vM(0,5)}$. For the noise in the covariates, $\varepsilon$ follows a centered wrapped Laplace with scale $\lambda_{\varepsilon} > 0$. The Fourier transform of the density of $\varepsilon$ is given by

$$ f_{\varepsilon}^{*}(l) = \frac{\lambda_{\varepsilon}^{2} e^{i0l}}{(l - i \frac{\lambda_{\varepsilon}}{\kappa_{\varepsilon}})(l + i \frac{\lambda_{\varepsilon}}{\kappa_{\varepsilon}})} = \frac{\lambda_{\varepsilon}^{2}}{l^{2} + \lambda_{\varepsilon}^{2}}, \quad \text{for } l \in \mathbb{Z}, \qquad (30) $$

giving an example of an ordinary smooth noise of degree $\nu = 2$ in (8). Figure 3 illustrates our regression setting.

We implement the estimator $\hat{m}_{\hat{L}}(x) = \text{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}}(x))$ (defined in (15)), where the levels $\hat{L}_1$ and $\hat{L}_2$ are selected via the procedure (13). To this end, we need to tune parameters $c_{0,1}$ and $c_{0,2}$. Similarly to the linear predictor case above, we implement preliminary simulations to calibrate them in the case $c_{0,1} = c_{0,2} = c_0$ by measuring the oracle risk $\mathcal{R}(c_0)$ given in (28). We consider the following grid:

$$ G_{c_0} := \{0.001; 0.0025; 0.005; 0.0075; 0.01; 0.02; 0.04; 0.05; 0.06; 0.08; 0.09; 0.1; 0.2; 0.4; 0.6; 1; 2; 4\}. $$

The function $\mathcal{R}(c_0)$ is plotted in Figure 4 when estimating at $x = 1.5$, for a covariate noise scale parameter $\lambda_{\varepsilon} = 2.54$, a sample size $n = 200$ and 50 Monte Carlo runs. A clear plateau phenomenon appears, showing that the choice of $c_{0,1} = c_{0,2} = 0.08$ is reasonable for further simulations.

We also would like to display the reliability ratio associated to each simulation. To do so, according to (29) we need to find the values of $\text{Var}(X)$ and $\text{Var}(\varepsilon)$ which are given below for a von
<figure><img src="image_5.png" /><figcaption>Figure 3: Illustration of model (CC) with $n = 200$. Simulated data $(\Theta_i)_{i=1}^n$ are displayed in green points. The red curve represents the regression function $m$, while the blue vertical line displays the point $x = 1.5$ where we aim at estimating $m(x)$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Plot of the function $c_0 \in G_{c_0} \to \mathcal{R}(c_0)$. The red vertical line displays the point $c_0 = 0.08$.</figcaption></figure>

Mises distribution $f_{vM(0,0.01)}$ and a centered wrapped Laplace distribution of scale parameter $\lambda_{\varepsilon}$, respectively:

$$ \mathrm{Var}(X) = 1 - \frac{I_1(0.01)}{I_0(0.01)}, $$

where $I_{\alpha}(\kappa)$ is the Bessel function of order $\alpha$ and

$$ \mathrm{Var}(\varepsilon) = 1 - \frac{\lambda_{\varepsilon}^2}{\sqrt{\left(\frac{1}{\kappa_{\varepsilon}^2} + \lambda_{\varepsilon}^2\right)\left(\kappa_{\varepsilon}^2 + \lambda_{\varepsilon}^2\right)}} = \frac{1}{1 + \lambda_{\varepsilon}^2}. \quad (31) $$

Finally, Table 2 displays our numerical performances showing again that they are again satisfying.

## 5.2 Real data application

We analyze the real data set described in Example 6.3 of Fisher's monograph *Statistical Analysis of Circular Data* [17], which arises from experiments on the distances ($X$) and directions ($\Theta$) moved by 31 small blue periwinkles (*Nodilittorina unifasciata*) after transplantation downshore from the
<table><caption>Table 2: Mean errors of $\hat{m}_{\hat{L}}(x)$ at point $x_0 = 1.5$</caption><thead><tr><th>λ<sub>ε</sub></th><th>Υ</th><th>n = 200</th><th>n = 500</th></tr></thead><tbody><tr><td>2.54</td><td>0.88</td><td>0.0124</td><td>0.0091</td></tr><tr><td>1.74</td><td>0.80</td><td>0.0139</td><td>0.0107</td></tr></tbody></table>

height at which they normally live. In fact, 15 observations correspond to individuals measured one day after transplantation, and the remaining 16 to those measured four days later. Note that no significant differences were observed between the behaviors of these two groups. Our aim is to predict direction from distance traveled, placing the problem in the framework of a regression model with a circular response and a linear predictor.

Di Marzio et al. [11] analyzed this data set using a nonparametric kernel-weighted regression estimator, while numerous parametric estimators have also been proposed: Fisher and Lee [18] considered the regression function, $\hat{m}_{\text{FL}}(x) = 1.693 + 2 \tan^{-1}(-0.0066(x - 47.65))$, while Presnell et al. [36] proposed $\hat{m}_{\text{SPML}}(x) = \text{atan2}(0.157 + 0.049x, -1.228 + 0.03x)$. More recently, Jammalamadaka and SenGupta [21] studied the alternative parametric model $\hat{m}_{\text{trig}}(x) = \text{atan2}(1 + 0.021x, -1.49 + 0.029x)$. To the best of our knowledge, this data set has not yet been analyzed within an errors-in-variables framework.

For this data set, we perform numerical experiments under two distinct scenarios in order to evaluate the performance of our data-driven statistical procedure. In the first scenario, we consider that the covariate is contaminated by an ordinary-smooth measurement error with Laplace distribution, $\varepsilon \sim \text{Laplace}(0, \sigma_{\varepsilon})$, where the scale parameter is set to $\sigma_{\varepsilon} = 0.1$. The bandwidths $h_i$ are selected using our procedure with tuning constants $c_{0,1} = c_{0,2} = 0.4$. In the second scenario, the covariate is observed with a supersmooth Gaussian measurement error, $\varepsilon \sim N(0, \sigma_{\varepsilon}^2)$, with $\sigma_{\varepsilon} = 0.1$.

<figure><img src="image_7.png" /><figcaption>Figure 5: The red curve displays reconstruction of the regression function $m$ in Scenario 1: $\varepsilon \sim \text{Laplace}(0, \sigma_{\varepsilon})$ with $\sigma_{\varepsilon} = 0.1$.</figcaption></figure>

The estimated curves are displayed in Figures 5 and 6 (red). For comparison, we also report the three parametric models proposed by Fisher and Lee [18], Presnell et al. [36], and Jammalamadaka and SenGupta [21]; note that the Fisher-Lee model is linear. Our nonparametric estimator GL, which explicitly accounts for measurement error in the covariates, appears somewhat undersmoothed, with
<figure><img src="image_8.png" /><figcaption>Figure 6: The red curve displays reconstruction of the regression function $m$ in Scenario 2: $\varepsilon \sim N(0, \sigma_{\varepsilon}^2)$ with $\sigma_{\varepsilon} = 0.1$.</figcaption></figure>

this effect becoming more pronounced under Gaussian noise. However, as noted by Presnell et al. [36], model comparison is not straightforward in the presence of heteroscedasticity.

# 6 Proofs

## 6.1 Circular predictor case

### 6.1.1 Proof of Proposition 3.2

*Proof.* First, let us upperbound the bias. Assume that $p_i \in W(\beta_i, S_i)$ with $\beta_i > \frac{1}{2}$. As $\mathbb{E}[\hat{p}_{i,L_i}(x)] = p_{i,L_i}(x)$ we have that $|\mathbb{E}[\hat{p}_{i,L_i}(x)] - p_i(x)| = |p_{i,L_i}(x) - p_i(x)|$. And we get

$$ 
\begin{aligned}
2\pi|p_i(x) - p_{i,L_i}(x)| &\le \sum_{|l| \ge L_i} |p_i^*(l)| = \sum_{|l| \ge L_i} |p_i^*(l)|(1+l^2)^{\beta_i/2}(1+l^2)^{-\beta_i/2} \\
&\le (\sum_{|l| \ge L_i} |p_i^*(l)|^2(1+l^2)^{\beta_i})^{1/2}(\sum_{|l| \ge L_i} (1+l^2)^{-\beta_i})^{1/2} \\
&\le C(S_i, \beta_i)L_i^{-\beta},
\end{aligned}
$$

where $C(S_i, \beta_i)$ is a constant depending on $S_i$ and $\beta_i$.
Let us prove the variance bounds. For $i \in \{1, 2\}$ we have

$$
\begin{align*}
(2\pi)^2 n \mathrm{Var}(\hat{p}_{i,L_i}(x)) &\le \mathbb{E} \left| \sum_{|l| \le L_i} e^{ilZ_1} e^{-ilx} \frac{1}{f_{\varepsilon}^*(l)} \right|^2 \\
&= \mathbb{E} \sum_{|l| \le L_i} e^{ilZ_1} e^{-ilx} \frac{1}{f_{\varepsilon}^*(l)} \sum_{|l'| \le L_i} e^{-il'Z_1} e^{il'x} \frac{1}{f_{\varepsilon}^*(l')} \\
&= \sum_{|l|, |l'| \le L_i} f_Z^*(l-l') e^{-i(l-l')x} \frac{1}{f_{\varepsilon}^*(l)} \frac{1}{f_{\varepsilon}^*(l')} \\
&\le \sum_{|l|, |l'| \le L_i} |f_Z^*(l-l')| \frac{1}{|f_{\varepsilon}^*(l)|} \frac{1}{|f_{\varepsilon}^*(l')|} = \sum_{|\tilde{l}|, |l| \le L_i} |f_Z^*(\tilde{l})| \frac{1}{|f_{\varepsilon}^*(l)|} \frac{1}{|f_{\varepsilon}^*(l-\tilde{l})|} \\
&\le \sum_{|\tilde{l}| \le L_i} |f_Z^*(\tilde{l})| \left( \sum_{|l| \le L_i} \frac{1}{|f_{\varepsilon}^*(l)|^2} \right)^{1/2} \left( \sum_{|l| \le L_i} \frac{1}{|f_{\varepsilon}^*(l-\tilde{l})|^2} \right)^{1/2} \\
&\le \sum_{|\tilde{l}| \le L_i} |f_X^*(\tilde{l})| |f_{\varepsilon}^*(\tilde{l})| \sum_{|l| \le 2L_i} \frac{1}{|f_{\varepsilon}^*(l)|^2} \le \sum_{|l| \le 2L_i} \frac{1}{|f_{\varepsilon}^*(l)|^2} \sum_{|l| \le L_i} \left| \int e^{ilx} f_X(x) dx \right| |f_{\varepsilon}^*(l)| \\
&\le \sum_{|l| \le 2L_i} \frac{1}{|f_{\varepsilon}^*(l)|^2} \sum_{|l| \le L_i} |f_{\varepsilon}^*(l)| \le \|f_{\varepsilon}^*\|_{\ell_1} \sum_{|l| \le 2L_i} \frac{1}{|f_{\varepsilon}^*(l)|^2}.
\end{align*}
$$

But we also have that

$$
(2\pi)^2 n \mathrm{Var}(\hat{p}_{i,L_i}(x)) \le \mathbb{E} \left| \sum_{|l| \le L_i} e^{ilZ_1} e^{-ilx} \frac{1}{|f_{\varepsilon}^*(l)|} \right|^2 \le \left( \sum_{|l| \le L_i} \frac{1}{|f_{\varepsilon}^*(l)|} \right)^2,
$$

hence

$$
\mathrm{Var}(\hat{p}_{i,L_i}(x)) \le \frac{1}{(2\pi)^2 n} \min \left( \left( \sum_{|l| \le L_i} \frac{1}{|f_{\varepsilon}^*(l)|} \right)^2, \|f_{\varepsilon}^*\|_{\ell_1} \sum_{|l| \le 2L_i} \frac{1}{|f_{\varepsilon}^*(l)|^2} \right).
$$

We now turn to the last point of the proposition which gives upper bounds for the variance when
noise assumptions are made.

If the noise is ordinary smooth, we have provided $\nu > 1$ (to ensure that $\|f_{\varepsilon}^{*}\|_{\ell_1}$ is finite) and
$L_i \ge 1$

$$
\mathrm{Var}(\hat{p}_{i,L_i}(x)) \le \frac{1}{(2\pi)^2 n} \max(1, \|f_{\varepsilon}^*\|_{\ell_1}) \min(L_i^{2\nu+2}, L_i^{2\nu+1}) \le \frac{1}{(2\pi)^2 n} \max(1, \|f_{\varepsilon}^*\|_{\ell_1}) L_i^{2\nu+1}.
$$

If the noise is supersmooth, then we get if $L_i \ge 1$

$$
\begin{align*}
\mathrm{Var}(\hat{p}_{i,L_i}(x)) &\le \frac{1}{(2\pi)^2 n} \max(1, \|f_{\varepsilon}^*\|_{\ell_1}) \min(L_i^{-2c+1} e^{2bL_i^a}, L_i^{-2c+2} e^{2bL_i^a}) \\
&= \frac{1}{(2\pi)^2 n} \max(1, \|f_{\varepsilon}^*\|_{\ell_1}) L_i^{-2c+1} e^{2bL_i^a},
\end{align*}
$$

which completes the proof of the proposition.

<figure><img src="image_9.png" /></figure>
### 6.1.2 Proof of Theorem 3.3

We first start with a proposition which gives some useful concentration results for our estimators.

**Proposition 6.1.** Suppose that the noise $\varepsilon$ is OS. For all $(L_i, L'_i) \in \mathcal{L}^2$, $i \in \{1, 2\}$ and for all $\alpha \ge 1$ we have that

$$ \mathbb{P}[|\hat{p}_{i,L_i}(x) - p_{i,L_i}(x)| \ge c_1(\alpha)\sqrt{V(n, L_i)}] \le 2n^{-\alpha} \quad (32) $$

$$ \mathbb{P}[|\hat{p}_{i,L_i \wedge L'_i}(x) - p_{i,L_i \wedge L'_i}(x)| \ge c_1(\alpha)\sqrt{V(n, L'_i)}] \le 2n^{-\alpha}, \quad (33) $$

as soon as $c_1^2(\alpha) \min(c_{0,1}, c_{0,2}) \ge \frac{16\alpha^2}{\min(\|f_\varepsilon^*\|_{\ell_1,1})}.$

*Proof.* To prove Proposition 6.1 we rely on the Bernstein inequality which is recalled in the next lemma.

**Lemma 6.2.** Let $T_1, \dots, T_n$ be n i.i.d variables and $S_n = \sum_{k=1}^n (T_k - \mathbb{E}(T_k))$. Then for any $\eta > 0$ we have

$$ \mathbb{P}(|S_n| \ge n\eta) \le 2\max \left\{ \exp\left(-\frac{n\eta^2}{4V}\right), \exp\left(-\frac{n\eta}{4b}\right) \right\}, \quad (34) $$

with $\mathrm{Var}(T_1) \le V$ and $|T_1| \le b$.

Now let us prove Proposition 6.1. We will prove it for $i = 1$, as the case $i = 2$ is strictly identical.

$$ \hat{p}_{1,L_1}(x) = \sum_{|l| \le L} \frac{1}{2\pi n} \sum_{j=1}^{n} \sin(\Theta_j) \frac{e^{ilZ_j}}{f_{\varepsilon}^{*}(l)} e^{-ilx}, x \in [-\pi, \pi). $$

Then setting

$$ T_{L_1,j}(x) := \frac{\sin(\Theta_j)}{2\pi} \sum_{|l| \le L_1} \frac{e^{ilZ_j}}{f_{\varepsilon}^{*}(l)} e^{-ilx}, $$

we have

$$ \hat{p}_{1,L_1}(x) = \frac{1}{n} \sum_{j=1}^{n} T_{L_1,j}(x). $$

We have that

$$ |T_{L_1,j}(x)| \le \frac{1}{2\pi} \sum_{|l| \le L_1} \frac{1}{|f_{\varepsilon}^{*}(l)|} =: b(L_1) \quad (35) $$

$$ \mathrm{Var}(T_{L_1,j}(x)) = n\mathrm{Var}(\hat{p}_{1,L_1}(x)) \le nV_0(n, L_1), \quad (36) $$

using Proposition 3.2 for the last inequality. Now applying Bernstein inequality to the variables $T_{L_1,j}(x)$, we get

$$ \begin{aligned} \mathbb{P}[|\hat{p}_{1,L_1}(x) - \mathbb{E}(\hat{p}_{1,L_1}(x))| \ge \eta(L_1)] &= \mathbb{P}[\sum_{j=1}^{n} T_{L_1,j}(x) - \mathbb{E}(T_{L_1,j}(x)) \ge n\eta(L_1)] \\ &\le 2\max \left\{ \exp\left(-\frac{n\eta^2(L_1)}{4nV_0(n, L_1)}\right), \exp\left(-\frac{n\eta(L_1)}{4b(L_1)}\right) \right\}. \end{aligned} $$

Let $\alpha \ge 1$ and choose $\eta(L_1) = c_1(\alpha)\sqrt{V(n, L_1)}$. Then we get

$$ \begin{aligned} & \mathbb{P}[|\hat{p}_{1,L_1}(x) - \mathbb{E}(\hat{p}_{1,L_1}(x))| \ge c_1(\alpha)\sqrt{V(n, L_1)}] \\ & \le 2\max \left\{ \exp\left(-\frac{nc_1^2(\alpha)V(n, L_1)}{4nV_0(n, L_1)}\right), \exp\left(-\frac{nc_1(\alpha)\sqrt{V(n, L_1)}}{4b(L_1)}\right) \right\}. \end{aligned} $$
Now let us choose $c_1(\alpha)$ in the following way. On the one hand, if $\frac{c_1^2(\alpha)c_{0,1}}{4} \ge \alpha$ then

$$ \frac{c_1^2(\alpha)V(n, L_1)}{4V_0(n, L_1)} = \frac{c_1^2(\alpha)}{4}c_{0,1} \log n \ge \alpha \log n. $$

On the other hand

$$ \frac{nc_1(\alpha)\sqrt{V(n, L_1)}}{4b(L_1)} = \frac{c_1(\alpha)\sqrt{c_{0,1} \log n}}{4} \frac{n\sqrt{V_0(n, L_1)}}{b(L_1)}. $$

But

$$ \frac{n\sqrt{V_0(n, L_1)}}{b(L_1)} = \sqrt{n} \min \left\{ 1, \frac{\|f_{\varepsilon}^{*}\|_{\ell_1}^{1/2}}{\sum_{|l| \le L_1} \frac{1}{|f_{\varepsilon}^{*}(l)|}} \left( \sum_{|l| \le 2L_1} \frac{1}{|f_{\varepsilon}^{*}(l)|^2} \right)^{1/2} \right\}. $$

Now since $L_1 \in \mathcal{L}$, this entails that

$$ \sqrt{n} \min \left\{ 1, \frac{\|f_{\varepsilon}^{*}\|_{\ell_1}^{1/2}}{\sum_{|l| \le L_1} \frac{1}{|f_{\varepsilon}^{*}(l)|}} \left( \sum_{|l| \le 2L_1} \frac{1}{|f_{\varepsilon}^{*}(l)|^2} \right)^{1/2} \right\} \ge \sqrt{c_2} \sqrt{\log n}, \quad (37) $$

with $\sqrt{c_2} = \min\{1, \|f_{\varepsilon}^{*}\|_{\ell_1}^{1/2}\}$, and we get that if $\frac{\sqrt{c_1^2(\alpha)c_{0,1}c_2}}{4} \ge \alpha$

$$ \frac{nc_1(\alpha)\sqrt{V(n, L_1)}}{4b(L_1)} \ge \alpha \log n. $$

Note that the condition $\frac{\sqrt{c_1^2(\alpha)c_{0,1}c_2}}{4} \ge \alpha$ implies the condition $\frac{c_1^2(\alpha)c_{0,1}}{4} \ge \alpha$, which completes the proof of the first inequality. Now for the second assertion of the proposition, we have

$$ \hat{p}_{1,L_1 \wedge L'_1}(x) = \frac{1}{n} \sum_{j=1}^{n} T_{L_1 \wedge L'_1, j}(x). \quad (38) $$

Hence, we have

$$ \begin{gathered} Var(\hat{p}_{L_1 \wedge L'_1}(x)) \le nV_0(n, L_1 \wedge L'_1) \le nV_0(n, L'_1) \\ |T_{L_1 \wedge L'_1, j}(x)| \le \frac{1}{2\pi} \sum_{|l| \le L_1 \wedge L'_1} \frac{1}{|f_{\varepsilon}^{*}(l)|} \le \sum_{|l| \le L'_1} \frac{1}{|f_{\varepsilon}^{*}(l)|} = b(L'_1), \end{gathered} $$

which entails the result following exactly the same line of reasoning as for the first assertion. This concludes the proof of Proposition 6.1. $\square$

Let us now prove Theorem 3.3.

*Proof.* Let $L \in \mathcal{L}_i$ be fixed and $i \in \{1, 2\}$.

$$ |\hat{p}_{i, \hat{L}_i}(x) - p_i(x)| \le \underbrace{|\hat{p}_{i, \hat{L}_i}(x) - \hat{p}_{i, L_i \wedge \hat{L}_i}(x)|}_{A_1} + \underbrace{|\hat{p}_{i, L_i \wedge \hat{L}_i}(x) - \hat{p}_{i, L_i}(x)|}_{A_2} + |\hat{p}_{i, L_i}(x) - p_i(x)|. \quad (39) $$

By definition of $A(L_i, x)$ (see 12), we have

$$ A_1 \le A(L_i, x) + \sqrt{V(n, \hat{L}_i)}. \quad (40) $$
By definition of $\hat{L}_i$ and $A(\hat{L}_i, x)$, we have

$$A_2 \leq A(\hat{L}_i, x) + \sqrt{V(n, L_i)}.$$

Thus

$$A_1 + A_2 \leq A(L_i, x) + \sqrt{V(n, \hat{L}_i)} + A(\hat{L}_i, x) + \sqrt{V(n, L_i)} \\ \leq 2A(L_i, x) + 2\sqrt{V(n, L_i)},$$

given the definition of $\hat{L}_i$. Thus

$$|\hat{p}_{i, \hat{L}_i}(x) - p_i(x)| \leq 2A(L_i, x) + 2\sqrt{V(n, L_i)} + |\hat{p}_{i, L_i}(x) - p_i(x)|.$$

Let us study more properly the term $A(L_i, x)$. We recall that

$$A(L_i, x) = \sup_{L'_i \in \mathcal{L}} \left\{ |\hat{p}_{i, L'_i}(x) - \hat{p}_{i, L_i \wedge L'_i}(x)| - \sqrt{V(n, L'_i)} \right\}_+$$

Now for any $L'_i \in \mathcal{L}$, we write

$$\hat{p}_{i, L'_i}(x) - \hat{p}_{i, L_i \wedge L'_i}(x) = (\hat{p}_{i, L'_i}(x) - p_{i, L'_i}(x)) - (\hat{p}_{i, L_i \wedge L'_i}(x) - p_{i, L_i \wedge L'_i}(x)) + p_{i, L'_i}(x) - p_{i, L_i \wedge L'_i}(x).$$

We have for the last term of the above decomposition

$$|p_{i, L'_i}(x) - p_{i, L_i \wedge L'_i}(x)| = \frac{1}{2\pi} \left| \sum_{|l| \le L'_i} p_i^*(l) e^{-ilx} - \sum_{|l| \le L_i \wedge L'_i} p_i^*(l) e^{-ilx} \right| \\ \le \frac{1}{2\pi} \sum_{|l| \ge L_i} |p_i^*(l)|.$$

Hence we get for $A(L_i, x)$ that

$$A(L_i, x) \leq \sup_{L'_i \in \mathcal{L}} [|\hat{p}_{i, L'_i}(x) - \hat{p}_{i, L_i \wedge L'_i}(x)| - \sqrt{V(n, L'_i)}]_+ \\ \leq \sup_{L'_i \in \mathcal{L}} [|\hat{p}_{i, L'_i}(x) - p_{i, L'_i}(x)| - \sqrt{V(n, L'_i)/2}]_+ \\ + \sup_{L'_i \in \mathcal{L}} [|\hat{p}_{i, L_i \wedge L'_i}(x) - p_{i, L'_i}(x)| - \sqrt{V(n, L'_i)/2}]_+ + \frac{1}{2\pi} \sum_{|l| \ge L_i} |p_i^*(l)|.$$

Using Proposition 6.1 with $c_1(\alpha) = \frac{1}{2}$, if $c_{0,i} \ge \frac{64\alpha^2}{\min(1, \|f_{\hat{L}_i}^*\|_{\ell_1})}$ we have

$$\mathbb{P}(\sup_{L'_i \in \mathcal{L}} [|\hat{p}_{i, L'_i}(x) - p_{i, L'_i}(x)| - \sqrt{V(n, L'_i)/2}]_+ > 0) \le 2 \sum_{L'_i \in \mathcal{L}} n^{-\alpha} \le 2n^{-\alpha+1},$$

as well as

$$\mathbb{P}(\sup_{L'_i \in \mathcal{L}} [|\hat{p}_{L_i \wedge L'_i}(x) - p_{i, L_i \wedge L'_i}(x)| - \sqrt{V(n, L'_i)/2}]_+ > 0) \le 2n^{-\alpha+1}.$$

This entails that if one considers the set

$$\mathcal{E}_i = \left\{ \sup_{L'_i \in \mathcal{L}} [|\hat{p}_{i, L'_i}(x) - p_{i, L'_i}(x)| - \sqrt{V(n, L'_i)/2}]_+ = 0 \right\} \\ \cap \left\{ \forall L_i \in \mathcal{L}, \sup_{L'_i \in \mathcal{L}} [|\hat{p}_{i, L_i \wedge L'_i}(x) - p_{i, L_i \wedge L'_i}(x)| - \sqrt{V(n, L'_i)/2}]_+ = 0 \right\},$$
one gets that $\mathbb{P}(\mathcal{E}_i) \ge 1 - 4n^{2-\alpha}$. Now let us choose $\alpha = 2 + q$. On the set $\mathcal{E}_i$ we get

$$
\begin{aligned}
|\hat{p}_{i,\hat{L}_i}(x) - p_i(x)| &\le 2A(L_i, x) + 2\sqrt{V(n, L_i)} + |\hat{p}_{i,L_i}(x) - p_i(x)| \\
&\le \frac{1}{2\pi} \sum_{|l|\ge L_i} |p_i^*(l)| + 2\sqrt{V(n, L_i)} + |\hat{p}_{i,L_i}(x) - p_{i,L_i}(x)| + |p_{i,L_i}(x) - p_i(x)| \\
&\le \frac{1}{2\pi} \sum_{|l|\ge L_i} |p_i^*(l)| + 2\sqrt{V(n, L_i)} + \sqrt{V(n, L_i)}/2 + \frac{1}{2\pi} \sum_{|l|\ge L_i} |p_i^*(l)| \\
&\le \frac{1}{\pi} \sum_{|l|\ge L_i} |p_i^*(l)| + \frac{5}{2}\sqrt{V(n, L_i)}.
\end{aligned}
$$

This concludes the proof of Theorem 3.3. $\square$

### 6.1.3 Proof of Theorem 3.4

We first need a lemma to bound the probability of a useful set.

**Lemma 6.3.** Let the noise be OS with $\nu > 1$. Let $i \in \{1, 2\}$ and $p_i$ belongs to $\mathcal{W}(\beta_i, S_i)$. Under the assumptions of Theorem 3.3, if one denotes

$$E_i := \left\{ |\hat{p}_{i,\hat{L}_i}(x) - p_i(x)|^2 \le C_i \left( \frac{\log n}{n} \right)^{\frac{2\beta_i}{2\beta_i+2\nu+1}} \right\},$$

with $C_i$ depending on $S_i, \beta_i, c_{0,i}$ and $f_{\varepsilon}^*$, then for $n$ large enough, we have

$$\mathbb{P}(E_i) \ge 1 - 4n^{-q}.$$

*Proof.* Let $i \in \{1, 2\}$. Using Proposition 3.2 which yields that $V(n, L_i) \lesssim \frac{\log(n)}{n} L_i^{2\nu+1}$ and that $\sum_{|l|\ge L_i} |p^*(l)| \lesssim L^{-\beta}$, one defines $L_{i,OS}$, the value $L_i$ such that

$$L_{i,OS} := \arg \min_{L_i \in \mathbb{N}^*} \left\{ L_i^{-2\beta} + \frac{\log(n)}{n} L_i^{2\nu+1} \right\}. \quad (41)$$

Let us prove that $L_{i,OS} \in \mathcal{L}$. Minimizing (41), one finds that $L_{i,OS}$ is proportional to $(\frac{n}{\log n})^{\frac{1}{2\beta_i+2\nu+1}}$. Now let $L \in \mathbb{N}^*$. We have that

$$\frac{\left(\sum_{|l| \le L} \frac{1}{|f_{\varepsilon}^*(l)|}\right)^2}{\sum_{|l| \le 2L} \frac{1}{|f_{\varepsilon}^*(l)|^2}} \lesssim \frac{\left(\sum_{|l| \le L} l^{\nu}\right)^2}{\sum_{|l| \le 2L} l^{2\nu}} \lesssim \frac{(L^{\nu+1})^2}{L^{2\nu+1}} \lesssim L.$$

Consequently, if $L \lesssim \frac{n}{\log n}$ then $L \in \mathcal{L}$. But, $L_{i,OS} \propto (\frac{n}{\log n})^{\frac{1}{2\beta_i+2\nu+1}} \le \frac{n}{\log n}$, hence $L_{i,OS} \in \mathcal{L}$. Consequently, using Theorem 3.3, we deduce that with probability greater than $1 - 4n^{-q}$

$$|\hat{p}_{i,\hat{L}_i}(x) - p_i(x)|^2 = \mathcal{O}\left(\frac{\log n}{n}\right)^{\frac{2\beta_i}{2\beta_i+2\nu+1}},$$

which completes the proof of Lemma 6.3. $\square$
We are now ready to prove Theorem 3.4. Using similar ideas to control $\mathbb{E}[d_c(\hat{m}_{\hat{L}}(x), m(x))]$ proposed in [34] we will first study the following term

$$R_n := \mathbb{E}[d_c(\hat{m}_{\hat{L}}(x), m(x)) \mathbf{1}_{E_1 \cap E_2}].$$

First, we have

$$R_n = \mathbb{E} \left[ \begin{aligned} & \left( 1 - \cos \left( \begin{aligned} & \operatorname{atan2}(\hat{p}_{1, \hat{L}_1}(x), \hat{p}_{2, \hat{L}_2}(x)) \\ & - \operatorname{atan2}(p_1(x), p_2(x)) \end{aligned} \right) \right) \mathbf{1}_{E_1 \cap E_2} \right] \\ &= 2\mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \begin{aligned} & \operatorname{atan2}(\hat{p}_{1, \hat{L}_1}(x), \hat{p}_{2, \hat{L}_2}(x)) \\ & - \operatorname{atan2}(p_1(x), p_2(x)) \end{aligned} \right) \right) \mathbf{1}_{E_1 \cap E_2} \right]. \end{aligned} \right.$$

Note that under Assumption (6) on the density $f_X$ of $X$, one has $m_j(x) = 0$ if and only if $p_j(x) = 0$. Now, we shall distinguish 3 cases.

**Case 1:** $|m_1(x)| > 0$ and $|m_2(x)| > 0$.

Let $\delta_1 = |m_1(x)|$ and $\delta_2 = |m_2(x)|$, meaning that $\delta = \min(\delta_1, \delta_2)$ (see (5)). First, on the event $E_1 \cap E_2$, for $n$ large enough satisfying

$$\psi_n(\beta_1, \nu) < \delta_1/2 \quad \text{and} \quad \psi_n(\beta_2, \nu) < \delta_2/2,$$

we have

$$|\hat{p}_{1, \hat{L}_1}(x) - p_1(x)| \le C_1 \psi_n(\beta_1, \nu) < \frac{|p_1(x)|}{2},$$

and

$$|\hat{p}_{2, \hat{L}_2}(x) - p_2(x)| \le C_2 \psi_n(\beta_2, \nu) < \frac{|p_2(x)|}{2}.$$

This implies that

$$\hat{p}_{1, \hat{L}_1}(x) p_1(x) > 0 \quad \text{and} \quad \hat{p}_{2, \hat{L}_2}(x) p_2(x) > 0,$$

and finally that $\hat{p}_{1, \hat{L}_1}(x)$ and $\hat{p}_{2, \hat{L}_2}(x)$ do not vanish on $E_1 \cap E_2$. Therefore,

$$\begin{aligned} R_n = 2\mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \arctan \left( \frac{\hat{p}_{1, \hat{L}_1}(x)}{\hat{p}_{2, \hat{L}_2}(x)} \right) - \arctan \left( \frac{p_1(x)}{p_2(x)} \right) \right) \right) \right. \\ \left. \times \mathbf{1}_{E_1 \cap E_2} \right] \\ \le \frac{1}{2} \mathbb{E} \left[ \left| \arctan \left( \frac{\hat{p}_{1, \hat{L}_1}(x)}{\hat{p}_{2, \hat{L}_2}(x)} \right) - \arctan \left( \frac{p_1(x)}{p_2(x)} \right) \right|^2 \mathbf{1}_{E_1 \cap E_2} \right] \\ \le \mathbb{E} \left[ \left| \arctan \left( \frac{\hat{p}_{1, \hat{L}_1}(x)}{\hat{p}_{2, \hat{L}_2}(x)} \right) - \arctan \left( \frac{p_1(x)}{\hat{p}_{2, \hat{L}_2}(x)} \right) \right|^2 \mathbf{1}_{E_1 \cap E_2} \right] \\ \left. + \mathbb{E} \left[ \left| \arctan \left( \frac{p_1(x)}{\hat{p}_{2, \hat{L}_2}(x)} \right) - \arctan \left( \frac{p_1(x)}{p_2(x)} \right) \right|^2 \mathbf{1}_{E_1 \cap E_2} \right]. \end{aligned} \quad (42)$$

On the event $E_2$, for $n$ sufficiently large,

$$|\hat{p}_{2, \hat{L}_2}(x)| \ge |p_2(x)| - |p_2(x) - \hat{p}_{2, \hat{L}_2}(x)| > \delta_2 - C_2 \psi_n(\beta_2, \nu) \ge \delta_2/2,$$
and using the 1-Lipschitz continuity of arctan, we get for the first term in (42), since on $E_1$ one has
$|\hat{p}_{1,\hat{L}_1}(x) - p_1(x)| \le C_1 \psi_n(\beta_1, \nu),$

$$
\begin{align*}
& \mathbb{E}\left[\left|\arctan\left(\frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)}\right) - \arctan\left(\frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)}\right)\right|^2 \mathbf{1}_{E_1 \cap E_2}\right] \\
& \le \frac{4}{\delta_2^2} \mathbb{E}\left[\left|\hat{p}_{1,\hat{L}_1}(x) - p_1(x)\right|^2 \mathbf{1}_{E_1 \cap E_2}\right] \\
& \le \frac{4}{\delta_2^2} \mathbb{E}\left[C_1 \psi_n^2(\beta_1, \nu) \mathbf{1}_{E_1 \cap E_2}\right] \\
& \le \frac{4}{\delta_2^2} C_1 \psi_n^2(\beta_1, \nu) \mathbb{P}(E_1 \cap E_2) \le \frac{4}{\delta_2^2} C_1 \psi_n^2(\beta_1, \nu).
\end{align*}
$$

Moreover, for the second term in (42), since $\frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)} \times \frac{p_1(x)}{p_2(x)} > 0$ on $E_2$, we have

$$
\begin{align*}
& \mathbb{E}\left[\left|\arctan\left(\frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)}\right) - \arctan\left(\frac{p_1(x)}{p_2(x)}\right)\right|^2 \mathbf{1}_{E_1 \cap E_2}\right] \\
&= \mathbb{E}\left[\left|\arctan\left(\frac{\hat{p}_{2,\hat{L}_2}(x)}{p_1(x)}\right) - \arctan\left(\frac{p_2(x)}{p_1(x)}\right)\right|^2 \mathbf{1}_{E_1 \cap E_2}\right] \\
&\le \frac{1}{|p_1(x)|^2} \mathbb{E}\left[\left|\hat{p}_{2,\hat{L}_2}(x) - p_2(x)\right|^2 \mathbf{1}_{E_1 \cap E_2}\right] \\
&\le \frac{1}{\delta_1^2} \mathbb{E}\left[C_2 \psi_n^2(\beta_2, \nu) \mathbf{1}_{E_1 \cap E_2}\right] \\
&\le \frac{1}{\delta_1^2} C_2 \psi_n^2(\beta_2, \nu).
\end{align*}
$$

Therefore, on the event $E_1 \cap E_2$, for $n$ sufficiently large such that

$$
C_1 \psi_n(\beta_1, \nu) \le \delta_1/2, \quad \text{and} \quad C_2 \psi_n(\beta_2, \nu) \le \delta_2/2,
$$

we obtain

$$
\mathbb{E}\left[\left|\hat{m}_{\hat{L}}(x) - m(x)\right|^2 \mathbf{1}_{E_1 \cap E_2}\right] \le \frac{4}{\delta^2} C_1 \psi_n^2(\beta_1, \nu) + \frac{1}{\delta^2} C_2 \psi_n^2(\beta_2, \nu).
$$

On the other hand, on the complementary $E_1^c \cup E_2^c$, using the fact that $|\operatorname{atan2}(w_1, w_2)| \le \pi$ for all $(w_1, w_2)$, we can simply obtain an upper-bound as follows:

$$
\begin{align*}
& \mathbb{E}\left[d_c(\hat{m}_{\hat{L}}(x) - m(x)) \mathbf{1}_{E_1^c \cup E_2^c}\right] \\
& \le \frac{1}{2} \mathbb{E}\left[\left|\operatorname{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}_2}(x)) - \operatorname{atan2}(p_1(x), p_2(x))\right|^2 \right. \\
& \qquad \left. \times \mathbf{1}_{E_1^c \cup E_2^c} \right] \\
& \le 2\pi^2 \mathbb{P}(E_1^c) + 2\pi^2 \mathbb{P}(E_2^c) \le 4\pi^2 4 n^{-q},
\end{align*}
$$

by Lemma 6.3. For $q \ge 1$, we get that $n^{-q}$ is negligible in comparison with $\psi_n^2(\beta_1, \nu)$ and $\psi_n^2(\beta_2, \nu)$.

**Case 2:** $m_1(x) = 0$ and $|m_2(x)| > 0$.

In this case $\delta = |m_2(x)|$.
- If $m_2(x) > 0$, then $p_2(x) > 0$ and as previously, on the event $E_1 \cap E_2$, for $n$ large enough, $\hat{p}_{2,\hat{L}_2}(x) > 0$. Then,

$$
\begin{align*}
R_n &= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}_2}(x)) - \operatorname{atan2}(p_1(x), p_2(x)) \right) \right) \right. \\
&\qquad \left. \times \mathbf{1}_{E_1 \cap E_2} \right] \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}_2}(x)) - 0 \right) \right) \mathbf{1}_{E_1 \cap E_2} \right] \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{arctan} \left( \frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) - \operatorname{arctan} \left( \frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) \right) \right) \right. \\
&\qquad \left. \times \mathbf{1}_{E_1 \cap E_2} \right] \\
&\le \frac{1}{2} \mathbb{E} \left[ \left| \operatorname{arctan} \left( \frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) - \operatorname{arctan} \left( \frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) \right|^2 \mathbf{1}_{E_1 \cap E_2} \right],
\end{align*}
$$

and we conclude as for the first case.

- If $m_2(x) < 0$, then $p_2(x) < 0$ and as previously, on the event $E_1 \cap E_2$, for $n$ large enough, $\hat{p}_{2,\hat{L}_2}(x) < 0$. Then,

$$
\begin{align*}
R_n &= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}_2}(x)) - \operatorname{atan2}(p_1(x), p_2(x)) \right) \right) \right. \\
&\qquad \left. \times \mathbf{1}_{E_1 \cap E_2} \right] \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}_2}(x)) + \pi \right) \right) \mathbf{1}_{E_1 \cap E_2} \right] \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{arctan} \left( \frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) + 2\pi \mathbf{1}_{\{\hat{p}_{1,\hat{L}_1}(x)>0\}} \right. \right. \\
&\qquad \left. \left. - \operatorname{arctan} \left( \frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) \right) \right) \mathbf{1}_{E_1 \cap E_2} \right] \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{arctan} \left( \frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) - \operatorname{arctan} \left( \frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) \right) \right) \right. \\
&\qquad \left. \times \mathbf{1}_{E_1 \cap E_2} \right] \\
&\le \frac{1}{2} \mathbb{E} \left[ \left| \operatorname{arctan} \left( \frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) - \operatorname{arctan} \left( \frac{p_1(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) \right|^2 \mathbf{1}_{E_1 \cap E_2} \right],
\end{align*}
$$

and we conclude as for the first case.

**Case 3:** $|m_1(x)| > 0$ and $m_2(x) = 0$.

In this case $\delta = |m_1(x)|$.

- If $m_1(x) > 0$, then $p_1(x) > 0$ and as previously, on the event $E_1 \cap E_2$, for $n$ large enough,
$\hat{p}_{1,\hat{L}_1}(x) > 0$. Then,

$$
\begin{align*}
R_n &= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}_2}(x)) - \operatorname{atan2}(p_1(x), p_2(x)) \right) \right) \right] \\
&\quad \times \mathbf{1}_{E_1 \cap E_2} \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{arctan} \left( \frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) + \pi \mathbf{1}_{\{\hat{p}_{2,\hat{L}_2}(x) < 0\}} - \frac{\pi}{2} \right) \right) \right] \\
&\quad \times \mathbf{1}_{E_1 \cap E_2} \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{arctan} \left( \frac{\hat{p}_{2,\hat{L}_2}(x)}{\hat{p}_{1,\hat{L}_1}(x)} \right) - \operatorname{arctan} \left( \frac{p_2(x)}{\hat{p}_{1,\hat{L}_1}(x)} \right) \right) \right) \right] \\
&\quad \times \mathbf{1}_{E_1 \cap E_2},
\end{align*}
$$

where the last equality is obtained by using for $u \in \mathbb{R} \setminus \{0\}$,

$$
\arctan(u) + \arctan(1/u) = \pi/2 \times \operatorname{sign}(u)
$$

and by distinguishing the cases according to the sign of $\hat{p}_{2,\hat{L}_2}(x)$.

- If $m_1(x) < 0$, then $p_1(x) < 0$ and as previously, on the event $E_1 \cap E_2$, for $n$ large enough, $\hat{p}_{1,\hat{L}_1}(x) < 0$. Then, similarly,

$$
\begin{align*}
R_n &= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{atan2}(\hat{p}_{1,\hat{L}_1}(x), \hat{p}_{2,\hat{L}_2}(x)) - \operatorname{atan2}(p_1(x), p_2(x)) \right) \right) \right] \\
&\quad \times \mathbf{1}_{E_1 \cap E_2} \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{arctan} \left( \frac{\hat{p}_{1,\hat{L}_1}(x)}{\hat{p}_{2,\hat{L}_2}(x)} \right) - \pi \mathbf{1}_{\{\hat{p}_{2,\hat{L}_2}(x) < 0\}} + \frac{\pi}{2} \right) \right) \right] \\
&\quad \times \mathbf{1}_{E_1 \cap E_2} \\
&= 2 \mathbb{E} \left[ \sin^2 \left( \frac{1}{2} \left( \operatorname{arctan} \left( \frac{\hat{p}_{2,\hat{L}_2}(x)}{\hat{p}_{1,\hat{L}_1}(x)} \right) - \operatorname{arctan} \left( \frac{p_2(x)}{\hat{p}_{1,\hat{L}_1}(x)} \right) \right) \right) \right] \\
&\quad \times \mathbf{1}_{E_1 \cap E_2}.
\end{align*}
$$

We conclude by using similar arguments presented for the second case since $\hat{p}_{1,\hat{L}_1}(x)$ (respectively $p_1(x)$) and $\hat{p}_{2,\hat{L}_2}(x)$ (respectively $p_2(x)$) play a symmetric role. Note that under Assumption (5), the case $m_1(x) = m_2(x) = 0$ cannot occur. This concludes the proof of the ordinary smooth noise case.

As for the supersmooth noise case, the proof is identical to that of the ordinary smooth case, and is in fact a much simpler version. Indeed, in the supersmooth setting, there is no need to select a bandwidth, and hence no need to consider the events $E_1$ and $E_2$. One simply follows the proof of the ordinary smooth case, omitting the indicator $\mathbf{1}_{E_1 \cap E_2}$ wherever it appears. This completes the proof of Theorem 3.4.

6.2 Linear predictor case

We now present proofs for the linear predictor case. Along this subsection, we fix $x \in [0, 1]$.
### 6.2.1 Proof of Proposition 4.6

We first state the following helpful lemma.

**Lemma 6.4.** (see [5, Lemma 1]) Let $c, s \ge 0$ and $\varrho \in \mathbb{R}$ such that $2\varrho > -1$ if $c = 0$ or $s = 0$. Then, for all $\omega > 0$, there exist constants $\lambda_1, \lambda_2 > 0$ such that

$$ \lambda_1 \omega^{2\varrho+1-s} \exp(c \omega^s) \le \int_{-\omega}^{\omega} (1+|u|^2)^{\varrho} \exp(c|u|^s) du \le \lambda_2 \omega^{2\varrho+1-s} \exp(c \omega^s). \quad (43) $$

Let us start proving Proposition 4.6. We will prove it for the case $j = 1$ as the other case $j = 2$ is strictly identical.

The upper bound for the bias of $\hat{p}_{1,h_1}(x)$ at $x \in [0, 1]$ is classical, for instance, one can find a proof in [34, Lemma 5.1]. Now, for the variance term, we have

$$ \mathrm{Var}(\hat{p}_{1,h_1}(x)) \le \frac{1}{(2\pi)^2 n} \mathbb{E} \left( \sin^2(\Theta_1) \times \left| \int_{\mathbb{R}} e^{-itx} e^{itZ_j} \frac{K_{h_1}^*(t)}{f_{\varepsilon}^*(t)} dt \right|^2 \right) \le \frac{1}{(2\pi)^2 n} \left\| \frac{K_{h_1}^*(t)}{f_{\varepsilon}^*(t)} \right\|_1^2. $$

On the other hand, we can also write

$$ \begin{aligned} (2\pi)^2 n \mathrm{Var}(\hat{p}_{1,h_1}(x)) &\le \mathbb{E} \left( \left| \int_{\mathbb{R}} e^{-itx} e^{itZ_1} \frac{K_{h_1}^*(t)}{f_{\varepsilon}^*(t)} dt \right|^2 \right) \\ &= \mathbb{E} \left( \left| \int_{\mathbb{R}} e^{-itx} e^{itZ_1} \frac{K_{h_1}^*(t)}{f_{\varepsilon}^*(t)} dt \right| \overline{\left| \int_{\mathbb{R}} e^{-itx} e^{itZ_1} \frac{K_{h_1}^*(t)}{f_{\varepsilon}^*(t)} dt \right|} \right) \\ &= \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-i(t-u)x} \int_{\mathbb{R}} e^{i(t-u)v} f_Z(v) dv \frac{K_{h_1}^*(t) K_{h_1}^*(-u)}{f_{\varepsilon}^*(t) f_{\varepsilon}^*(-u)} dt du \\ &= \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-i(t-u)x} f_Z^*(t-u) \frac{K_{h_1}^*(t) K_{h_1}^*(-u)}{f_{\varepsilon}^*(t) f_{\varepsilon}^*(-u)} dt du \\ &\le \int_{\mathbb{R}} \left( \int_{\mathbb{R}} \left| \frac{K_{h_1}^*(v+u) K_{h_1}^*(-u)}{f_{\varepsilon}^*(v+u) f_{\varepsilon}^*(-u)} \right| du \right) |f_Z^*(v)| dv \\ &\le \int_{\mathbb{R}} \left( \int_{\mathbb{R}} \left| \frac{K_{h_1}^*(v+u)}{f_{\varepsilon}^*(v+u)} \right|^2 du \right)^{\frac{1}{2}} \left( \int_{\mathbb{R}} \left| \frac{K_{h_1}^*(-u)}{f_{\varepsilon}^*(-u)} \right|^2 du \right)^{\frac{1}{2}} |f_Z^*(v)| dv \\ &\le \left\| \frac{K_{h_1}^*}{f_{\varepsilon}^*} \right\|_2^2 \int_{\mathbb{R}} |f_X^*(v) f_{\varepsilon}^*(v)| dv = \left\| \frac{K_{h_1}^*}{f_{\varepsilon}^*} \right\|_2^2 \int_{\mathbb{R}} \left( \left| \int_{\mathbb{R}} e^{ivy} f_X(y) dy \right| \right) |f_{\varepsilon}^*(v)| dv \\ &\le \left\| \frac{K_{h_1}^*}{f_{\varepsilon}^*} \right\|_2^2 \|f_{\varepsilon}^*\|_1, \end{aligned} $$

as $|e^{ivy}| \le 1$ for any $v, y \in \mathbb{R}$ and $\int_{\mathbb{R}} f_X(y) dy = 1$. This leads to the fact that, for any $x \in [0, 1]$,

$$ \mathrm{Var}(\hat{p}_{1,h_1}(x)) \le \frac{1}{(2\pi)^2 n} \min \left\{ \left\| \frac{K_{h_1}^*}{f_{\varepsilon}^*} \right\|_2^2 \|f_{\varepsilon}^*\|_1, \left\| \frac{K_{h_1}^*}{f_{\varepsilon}^*} \right\|_1^2 \right\} = \tilde{V}_0(n, h_1). $$

Now, for the ordinary-smooth noise, under Assumption (16) with $r > 1$ and the constant $c_\varepsilon > 0$, one has

$$ (2\pi)^2 n V_0(n, h_1) \le c_\varepsilon^{-2} \min \left\{ \|f_\varepsilon^*\|_1 \int_{\mathbb{R}} |K_{h_1}^*(t)|^2 (1+|t|)^{2r} dt, \left( \int_{\mathbb{R}} |K_{h_1}^*(t)| (1+|t|)^r dt \right)^2 \right\}. $$
Moreover, recall that $K_{h_1}^*(t) = K^*(t h_1)$ for any $t \in \mathbb{R}$, we have

$$
\begin{align*}
\|f_{\varepsilon}^{*}\|_{1} \int_{\mathbb{R}} |K_{h_1}^{*}(t)|^{2} (1 + |t|)^{2r} dt &= \|f_{\varepsilon}^{*}\|_{1} \int_{\mathbb{R}} |K^{*}(t h_1)|^{2} (1 + |t|)^{2r} dt \\
&= \|f_{\varepsilon}^{*}\|_{1} h_1^{-1} \int_{\mathbb{R}} |K^{*}(u)|^{2} (1 + \left|\frac{u}{h_1}\right|)^{2r} du \\
&\leq \|f_{\varepsilon}^{*}\|_{1} h_1^{-1-2r} \int_{\mathbb{R}} |K^{*}(u)|^{2} (1 + |u|)^{2r} du \\
&\leq \|f_{\varepsilon}^{*}\|_{1} h_1^{-1-2r} C_{(K^{*})},
\end{align*}
$$

and

$$
\left( \int_{\mathbb{R}} |K_{h_1}^*(t)| (1+|t|)^r dt \right)^2 \leq (h_1^{-1-r})^2 \left( \int_{\mathbb{R}} |K^*(u)| (1+|u|)^r du \right)^2 \leq h_1^{-2-2r} C_{(K^*)},
$$

where $C_{(K^*)}$ is defined in Assumption 4.5. Thus, if $0 < h_1 \le 1$ we obtain,

$$
\begin{align*}
V_0(n, h_1) &\le \frac{c_\epsilon^{-2} C_{(K^*)} \min \{\|f_\epsilon^*\|_1, h_1^{-1-2r}, h_1^{-2-2r}\}}{(2\pi)^2} n^{-1} \\
&\le \frac{c_\epsilon^{-2} C_{(K^*)} \max \{\|f_\epsilon^*\|, 1\} \times \min \{h_1^{-1-2r}, h_1^{-2-2r}\}}{(2\pi)^2} n^{-1} \\
&\le \frac{c_\epsilon^{-2} C_{(K^*)} \max \{\|f_\epsilon^*\|, 1\}}{(2\pi)^2} n^{-1} h_1^{-1-2r}.
\end{align*}
$$

For the super-smooth noise, under Assumption (17) one has

$$
(2\pi)^2 n V_0(n, h_1) \\
\le \tilde{c}_{\epsilon}^{-2} \min \left\{ \|f_{\epsilon}^{*}\|_{1} \int_{\mathbb{R}} |K_{h_1}^{*}(t)|^2 (1 + |t|^2)^{\rho_0} \exp(2\gamma|t|^{\rho}) dt, \left( \int_{\mathbb{R}} |K_{h_1}^{*}(t)| (1 + |t|^2)^{\frac{\rho_0}{2}} \exp(\gamma|t|^{\rho}) dt \right)^2 \right\}.
$$

Using the sinc kernel for which we have $K^*(t) = \mathbf{1}_{[-1,1]}(t)$ for $t \in \mathbb{R}$, Lemma 6.4 with $\lambda_2 > 0$ (and
similar computations as in [5] page 589), one gets

$$
\|f_{\varepsilon}^{*}\|_{1} \int_{\mathbb{R}} |K_{h_1}^{*}(t)|^2 (1 + |t|^2)^{\rho_0} \exp(2\gamma|t|^{\rho}) dt \leq \lambda_2 \|f_{\varepsilon}^{*}\|_{1} h_1^{-2\rho_0-1+\rho} \exp(2\gamma h_1^{-\rho}).
$$

On the other hand, we have

$$
\left( \int_{\mathbb{R}} |K_{h_1}^*(t)| (1 + |t|^2)^{\frac{\rho_0}{2}} \exp(\gamma |t|^\rho) dt \right)^2 \le \lambda_2^2 h_1^{-2\rho_0 - 2 + 2\rho} \exp(2\gamma h_1^{-\rho}).
$$

Hence, we obtain the following upper-bound for $V_0(n, h_1)$ in the super-smooth noise case,

$$
\begin{align*}
V_0(n, h_1) &\le \frac{(\lambda_2 + \lambda_2^2) \tilde{c}_\epsilon^{-2} \max \{\|f_\epsilon^*\|; 1\}}{(2\pi)^2} n^{-1} \min \{1; h_1^{-1+\rho}\} h_1^{-2\rho_0-1+\rho} \exp (2\gamma h_1^{-\rho}) \\
&= \frac{(\lambda_2 + \lambda_2^2) \tilde{c}_\epsilon^{-2} \max \{\|f_\epsilon^*\|; 1\}}{(2\pi)^2} n^{-1} h_1^{(\rho-1)+} h_1^{-2\rho_0-1+\rho} \exp (2\gamma h_1^{-\rho}).
\end{align*}
$$

Similar arguments apply for $\mathrm{Var}(\hat{p}_{2,h_2}(x))$ and we conclude the proof of Proposition 4.6.
### 6.2.2 Proof of Theorem 4.7

First, we will establish some concentration results for $\hat{p}_{j,h_j}$ with $j \in \{1, 2\}$ in the following proposition.

**Proposition 6.5.** Let $j \in \{1, 2\}$. Under the Assumptions of Theorem 4.7, for all $(h_j, h'_j) \in \mathcal{H}^2$, for all $\alpha \ge 1$,

$$ \mathbb{P}\left(\left|\hat{p}_{j,h_j}(x) - \mathbb{E}[\hat{p}_{j,h_j}(x)]\right| > \tilde{c}_1(\alpha)\sqrt{\tilde{V}(n, h'_j)}\right) \le 2n^{-\alpha}, $$

$$ \mathbb{P}\left(\left|\hat{p}_{j,h_j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)]\right| > \tilde{c}_1(\alpha) \|K^\star\|_{\infty} \sqrt{\tilde{V}(n, h'_j)}\right) \le 2n^{-\alpha}, $$

as soon as $\tilde{c}_1(\alpha)$ satisfying $\tilde{c}_1^2(\alpha) \min\{\tilde{c}_{0;1}; \tilde{c}_{0;2}\} \ge \frac{16\alpha^2}{\min\{\|f_{\tilde{\epsilon}}^\star\|_{1,1}\}}$.

*Proof.* Let us prove Proposition 6.5. We only consider the case $j = 1$ as the case $j = 2$ is identical. For $1 \le k \le n$, we define the random variables

$$ U_{k,1}(x) := \sin(\Theta_k) \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} e^{itZ_k} \frac{K_{h_j}^*(t)}{f_{\tilde{\epsilon}}^*(t)} dt, $$

such that, $\hat{p}_{1,h_1}(x) = \frac{1}{n} \sum_{k=1}^n U_{k,1}(x)$. Notice that $\mathbb{E}(U_{k,1}(x)) = (K_{h_1} * p_1)(x)$. We have for any $x \in [0, 1]$,

$$ |U_{k,1}(x)| = \left| \sin(\Theta_k) \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} e^{itZ_k} \frac{K_{h_1}^*(t)}{f_{\tilde{\epsilon}}^*(t)} dt \right| \le \frac{1}{2\pi} \int_{\mathbb{R}} \left| \frac{K_{h_j}^*(t)}{f_{\tilde{\epsilon}}^*(t)} \right| dt = \frac{1}{2\pi} \left\| \frac{K_{h_j}^*}{f_{\tilde{\epsilon}}^*} \right\|_1 =: b(h_1), \quad (44) $$

and $\mathrm{Var}(U_{k,1}(x)) = n\mathrm{Var}(\hat{p}_{1,h_1}(x)) \le n\tilde{V}_0(n, h_1)$. Now, applying Lemma 6.2 to the $U_{k,1}$'s, we obtain for $\eta(h_1) > 0$,

$$ \mathbb{P}\left(\left|\hat{p}_{1,h_1}(x) - \mathbb{E}(\hat{p}_{1,h_1}(x))\right| \ge \eta(h_1)\right) = \mathbb{P}\left(\left|\sum_{k=1}^n U_{k,1}(x) - \mathbb{E}(U_{k,1}(x))\right| \ge n\eta(h_1)\right) \\ \le 2\max\left\{\exp\left(-\frac{n \times (\eta(h_1))^2}{4n\tilde{V}_0(n, h_1)}\right); \exp\left(-\frac{n \times \eta(h_1)}{4b(h_1)}\right)\right\}. $$

For $\alpha \ge 1$, choose $\eta(h_1) = \tilde{c}_1(\alpha)\sqrt{\tilde{V}(n, h_1)}$, with $\tilde{V}(n, h_1) = \tilde{c}_{0,1} \log(n)\tilde{V}_0(n, h_1)$. Then,

$$ \mathbb{P}\left(\left|\hat{p}_{1,h_1}(x) - \mathbb{E}(\hat{p}_{1,h_1}(x))\right| \ge \tilde{c}_1(\alpha)\sqrt{\tilde{V}(n, h_1)}\right) \\ \le 2\max\left\{\exp\left(-\frac{n \tilde{c}_1^2(\alpha)\tilde{V}(n, h_1)}{4n\tilde{V}_0(n, h_1)}\right), \exp\left(-\frac{n \tilde{c}_1(\alpha)\sqrt{\tilde{V}_1(n, h_1)}}{4b(h_1)}\right)\right\}. \quad (45) $$

We then choose $\tilde{c}_1(\alpha)$. First, $\tilde{c}_1(\alpha)$ is chosen such that

$$ \frac{n \tilde{c}_1^2(\alpha) \tilde{V}(n, h_1)}{4n \tilde{V}_0(n, h_1)} = \frac{n \tilde{c}_1^2(\alpha) \tilde{c}_{0,1} \log(n) \tilde{V}_0(n, h_1)}{4n \tilde{V}_0(n, h_1)} \ge \alpha \log(n), \quad (46) $$

that is $\tilde{c}_1(\alpha)$ satisfies $\tilde{c}_1^2(\alpha) \tilde{c}_{0,1} \ge 4\alpha$.

Secondly, we can write

$$ \frac{n \tilde{c}_1(\alpha) \sqrt{\tilde{V}_1(n, h_1)}}{4b(h_1)} = \frac{\tilde{c}_1(\alpha) \sqrt{\tilde{c}_{0,1}}}{4} \sqrt{\log(n)} \frac{n \sqrt{\tilde{V}_0(n, h_1)}}{b(h_1)}, $$
and for $h_1 \in \mathcal{H}$,

$$ 
\begin{aligned}
n \frac{\sqrt{V_0(n, h_1)}}{b(h_1)} &= n \frac{1}{2\pi\sqrt{n}} \min \left\{ \left\| \frac{K_{h_1}^*}{f_\varepsilon^*} \right\|_2 \|f_\varepsilon^*\|_1^{1/2} ; \left\| \frac{K_{h_1}^*}{f_\varepsilon^*} \right\|_1 \right\} \times 2\pi \left( \left\| \frac{K_{h_1}^*}{f_\varepsilon^*} \right\|_1 \right)^{-1} \\
&= \sqrt{n} \min \left\{ \left\| \frac{K_{h_1}^*}{f_\varepsilon^*} \right\|_2 \|f_\varepsilon^*\|_1^{1/2} \left( \left\| \frac{K_{h_1}^*}{f_\varepsilon^*} \right\|_1 \right)^{-1} ; 1 \right\} \geq \sqrt{c_3} \sqrt{\log(n)},
\end{aligned}
$$

with $\sqrt{c_3} = \min \{ \|f_\varepsilon^*\|_1^{1/2} ; 1 \}$, then we get

$$ \frac{n \tilde{c}_1(\alpha) \sqrt{\tilde{V}_1(n, h_1)}}{4b(h_1)} = \frac{\tilde{c}_1(\alpha) \sqrt{\tilde{c}_{0,1}}}{4} \sqrt{\log(n)} \frac{n \sqrt{V_0(n, h_1)}}{b(h_1)} \geq \alpha \log(n) \quad (47) $$

provided that $\tilde{c}_1(\alpha)\sqrt{\tilde{c}_{0,1}c_3} \geq 4\nu$. Note that this condition also ensures the constraint $\tilde{c}_1^2(\alpha)\tilde{c}_{0,1} \geq 4\alpha$. Now, combining (46) and (47), we get from (45) for any $\alpha \geq 1$

$$ \mathbb{P} \left( \left| \hat{p}_{1,h_1}(x) - \mathbb{E}(\hat{p}_{1,h_1}(x)) \right| > \tilde{c}_1(\alpha) \sqrt{\tilde{V}(n, h_1)} \right) \leq 2n^{-\alpha}. $$

Now, for the second assertion of Proposition 6.5, for $1 \leq k \leq n$, we define random variables

$$ \tilde{U}_{k,1}(x) := \sin(\Theta_k) \frac{1}{2\pi} \int_{\mathbb{R}} e^{-itx} e^{itZ_k} \frac{K_{h_1}^*(t) K_{h_1}^*(t)}{f_\varepsilon^*(t)} dt, $$

hence, $\hat{p}_{1,h_1,h_1'}(x) = \frac{1}{n} \sum_{k=1}^n \tilde{U}_{k,1}(x)$. We have

$$ \left| \tilde{U}_{k,1}(x) \right| \leq \frac{1}{2\pi} \int_{\mathbb{R}} \left| \frac{K_{h_1}^*(t) K_{h_1}^*(t)}{f_\varepsilon^*(t)} \right| dt \leq \frac{1}{2\pi} \|K^*\|_{\infty} \left\| \frac{K_{h_1}^*}{f_\varepsilon^*} \right\|_1, $$

and

$$ \mathrm{Var}(\tilde{U}_{k,1}(x)) = n \mathrm{Var}(\hat{p}_{1,h_1,h_1'}(x)) \leq n \|K^*\|_{\infty} \tilde{V}_0(n, h_1). $$

Using similar arguments in proof of the first assertion on $\hat{p}_{1,h_1}(x)$, we obtain with a probability greater than $1 - 2n^{-\alpha}$ that

$$ \left| \hat{p}_{1,h_1,h_1'}(x) - \mathbb{E}[\hat{p}_{1,h_1,h_1'}(x)] \right| \leq \tilde{c}_1(\alpha) \|K^*\|_{\infty} \sqrt{\tilde{V}(n, h_1)}. $$

This concludes the proof of Proposition 6.5. $\square$

Now, we can start to prove Theorem 4.7. As the proof is similar to the proof of Theorem 3.3, we will omit some computations.

Let $j \in \{1, 2\}$ and $h_j \in \mathcal{H}$ be fixed. Recall that the target is to find an upper bound for $|\hat{p}_{j,\hat{h}_j}(x) - p_j(x)|$. We consider the following decomposition:

$$ |\hat{p}_{j,\hat{h}_j}(x) - p_j(x)| \leq |\hat{p}_{j,\hat{h}_j}(x) - \hat{p}_{j,h_j,\hat{h}_j}(x)| + |\hat{p}_{j,h_j,\hat{h}_j}(x) - \hat{p}_{j,h_j}(x)| + |\hat{p}_{j,h_j}(x) - p_j(x)|. $$

Following proof of Theorem 3.3, we obtain

$$ |\hat{p}_{j,\hat{h}_j}(x) - p_j(x)| \leq 2\tilde{A}(h_j, x) + 2\sqrt{\tilde{V}_j(n, h_j)} + |\hat{p}_{j,h_j}(x) - p_j(x)|. \quad (48) $$
For the study of the term $\tilde{A}(h_j, x)$, we recall that:

$$ \tilde{A}(h_j, x) = \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h'_j}(x) - \hat{p}_{j,h_j,h'_j}(x) \right| - \sqrt{\tilde{V}_j(n, h'_j)} \right\}_+ . $$

We get that

$$ \begin{aligned} \left| \hat{p}_{j,h'_j}(x) - \hat{p}_{j,h_j,h'_j}(x) \right| - \sqrt{\tilde{V}_j(n, h'_j)} \le \left| \hat{p}_{j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h'_j}(x)] \right| - \frac{\sqrt{\tilde{V}_j(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \\ + \left| \hat{p}_{j,h_j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)] \right| - \|K^*\|_{\infty} \frac{\sqrt{\tilde{V}_j(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \\ + \left| \mathbb{E}[\hat{p}_{j,h'_j}(x)] - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)] \right|. \end{aligned} $$

However, for any $h'_j \in \mathcal{H}$, with $M(K)$ defined in (18), we get

$$ \left| \mathbb{E}[\hat{p}_{j,h'_j}(x)] - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)] \right| = \left| K_{h'_j} * (p_j - K_{h_j} * p_j)(x) \right| \le M(K) \left\| p_j - K_{h_j} * p_j \right\|_{\infty}. \quad (49) $$

Hence, incorporating this bound in the definition of $\tilde{A}(h_j, x)$, we obtain

$$ \begin{aligned} \tilde{A}(h_j, x) = \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h'_j}(x) - \hat{p}_{j,h_j,h'_j}(x) \right| - \sqrt{\tilde{V}(n, h'_j)} \right\}_+ \\ \le \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h'_j}(x)] \right| - \frac{\sqrt{\tilde{V}(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ \\ + \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h_j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)] \right| - \|K^*\|_{\infty} \frac{\sqrt{\tilde{V}(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ \\ + \|K^*\|_{\infty} \|p_j - K_{h_j} * p_j\|_{\infty} \\ \le \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h'_j}(x)] \right| - \frac{\sqrt{\tilde{V}(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ \\ + \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h_j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)] \right| - \|K^*\|_{\infty} \frac{\sqrt{\tilde{V}(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ \\ + M(K) \|p_j - K_{h_j} * p_j\|_{\infty}. \end{aligned} \quad (50) $$

By Proposition 6.5

$$ \mathbb{P} \left( \left| \hat{p}_{j,h_j}(x) - \mathbb{E}[\hat{p}_{j,h_j}(x)] \right| > c_1(\alpha) \sqrt{\tilde{V}(n, h_j)} \right) \le 2n^{-\alpha}. $$

It implies that if we take $c_1(\alpha) = \frac{1}{1 + \|K^*\|_{\infty}}$ and if $c_{0,j} \ge 16\alpha^2(1 + \|K^*\|_{\infty})^2 \frac{1}{\min\{\|f^*_\epsilon\|_1, 1\}}$, then

$$ \mathbb{P} \left( \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h'_j}(x)] \right| - \frac{\sqrt{\tilde{V}(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ > 0 \right) \le 2 \sum_{h_j \in \mathcal{H}} n^{-\alpha} \le 2n^{1-\alpha}, $$
as $\text{Card}(\mathcal{H}) \le n$ (see (23)). Similarly, from Proposition 6.5, we can deduce that

$$ \mathbb{P} \left( \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h_j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)] \right| - \|K^*\|_{\infty} \frac{\sqrt{\tilde{V}_j(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ > 0 \right) \le 2n^{1-\alpha}. $$

Consequently, the following set

$$ \tilde{\mathcal{E}}_j := \left\{ \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h'_j}(x)] \right| - \frac{\sqrt{\tilde{V}_j(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ = 0 \right\} \\ \cap \left\{ \forall h_j \in \mathcal{H}, \sup_{h'_j \in \mathcal{H}} \left\{ \left| \hat{p}_{j,h_j,h'_j}(x) - \mathbb{E}[\hat{p}_{j,h_j,h'_j}(x)] \right| - \|K^*\|_{\infty} \frac{\sqrt{\tilde{V}_j(n, h'_j)}}{(1 + \|K^*\|_{\infty})} \right\}_+ = 0 \right\} $$

has probability greater than $1 - 4n^{2-\alpha}$.

Now, for $\alpha \ge 3$, choose $\alpha = 2+q$ and then $\tilde{c}_{0,j} \ge 16(2+q)^2(1+\|K^*\|_{\infty})^2 \frac{1}{\min\{\|f_{\tilde{\mathcal{E}}_j}^*\|_1, 1\}}$. Thus, we obtain that $\mathbb{P}(\tilde{\mathcal{E}}_j) > 1 - 4n^{-q}$. Combining (48) and (50), we have on $\tilde{\mathcal{E}}_j$:

$$ \begin{aligned} |\hat{p}_{j,\hat{h}_j}(x) - p_j(x)| &\le 2\tilde{A}(h_j, x) + 2\sqrt{\tilde{V}_j(n, h_j)} + |\hat{p}_{j,h_j}(x) - p_j(x)| \\ &\le 2M(K) \|p_j - K_{h_j} * p_j\|_{\infty} + 2\sqrt{\tilde{V}_j(n, h_j)} + |\hat{p}_{j,h_j}(x) - p_j(x)|, \end{aligned} $$

but still on $\tilde{\mathcal{E}}_j$, one gets $|\hat{p}_{j,h_j}(x) - \mathbb{E}[\hat{p}_{j,h_j}(x)]| - \frac{\sqrt{\tilde{V}_j(n, h_j)}}{(1 + \|K^*\|_{\infty})} \le 0$, so

$$ \begin{aligned} |\hat{p}_{j,h_j}(x) - p_j(x)| &\le |\mathbb{E}[\hat{p}_{j,h_j}(x)] - p_j(x)| + |\hat{p}_{j,h_j}(x) - \mathbb{E}[\hat{p}_{j,h_j}(x)]| \\ &\quad - \frac{\sqrt{\tilde{V}_j(n, h_j)}}{(1 + \|K^*\|_{\infty})} + \frac{\sqrt{\tilde{V}_j(n, h_j)}}{(1 + \|K^*\|_{\infty})} \\ &\le \|p_j - K_{h_j} * p_j\|_{\infty} + \sqrt{\tilde{V}_j(n, h_j)}. \end{aligned} $$

Therefore, on $\tilde{\mathcal{E}}_j$, we finally obtain

$$ |\hat{p}_{j,\hat{h}_j}(x) - p_j(x)| \le (1 + 2M(K)) \|p_j - K_{h_j} * p_j\|_{\infty} + 3\sqrt{\tilde{V}_j(n, h_j)}. $$

This concludes the proof of Theorem 4.7.

### 6.2.3 Proof of Theorem 4.8

First, we establish a concentration result for $\hat{p}_{j,\hat{h}_j}(x)$ in the following lemma.

**Lemma 6.6.** Suppose that the noise $\varepsilon$ is OS with $r > 1$. Let $j \in \{1, 2\}$ and $p_j$ belong to $\mathcal{H}(\beta_j, \Lambda_j)$. Assume that the kernel $K$ satisfies Assumption 4.5 and Assumption 4.4 with an index $\kappa \in \mathbb{R}_+$ such that $\kappa \ge \beta_j$. Then under assumptions of Theorem 4.7, there exist constants $C_j$ depending on $\beta_j, \Lambda_j, \tilde{c}_{0,j}, K$ and $f_{\tilde{\mathcal{E}}_j}^*$ such that, with

$$ \tilde{E}_j := \left\{ |\hat{p}_{j,\hat{h}_j}(x) - p_j(x)| \le C_j \left( \frac{\log(n)}{n} \right)^{\frac{2\beta_j}{2\beta_j+2r+1}} \right\}, $$

we have, for $n$ large enough,

$$ \mathbb{P}(\tilde{E}_j) > 1 - 4n^{-q}. $$
Proof. Let $j \in \{1, 2\}$. Since $p_j$ belongs to $\mathcal{H}(\beta_j, \Lambda_j)$, from Proposition 4.6, we have

$$ \left| \mathbb{E}(\hat{p}_{j,h_j}(x)) - p_j(x) \right| \le C_{K,\kappa} \Lambda_j h_j^{\beta_j}, $$

and

$$ \tilde{V}(n, h_j) \lesssim \frac{\log n}{n} h_j^{-1-2r}. $$

Thus, we define $h_{j,OS}$ by

$$ h_{j,OS} := \arg \min_{h_j \in \{k^{-1}: k \in \mathbb{N}^*\}} \left\{ h_j^{2\beta_j} + \frac{\log n}{n} h_j^{-1-2r} \right\}. \quad (51) $$

Equation (51) yields that $h_{j,OS}$ is of order $(\log(n)/n)^{1/(2\beta_j+1+2r)}$. Let us show that $h_{j,OS} \in \mathcal{H}$.
Let $0 \le h \le 1$. We have (using similar computations as in [5] page 598)

$$ \begin{aligned} \left\| \frac{K_h^*}{f_\varepsilon^*} \right\|_2^2 \left( \left\| \frac{K_h^*}{f_\varepsilon^*} \right\|_1 \right)^{-2} &= \int_{\mathbb{R}} \left| \frac{K^*(t h)}{f_\varepsilon^*(t)} \right|^2 dt \left( \int_{\mathbb{R}} \left| \frac{K^*(t h)}{f_\varepsilon^*(t)} \right| dt \right)^{-2} \\ &\geq h c_\varepsilon^{-2} C_\varepsilon^{-2} \int_{\mathbb{R}} |K^*(u)|^2 |u|^{2r} du \left( \int_{\mathbb{R}} |K^*(v)| (1+|v|)^r dv \right)^{-2}. \end{aligned} $$

Consequently,

$$ \begin{aligned} \left\| \frac{K_{h_{j,OS}}^*}{f_\varepsilon^*} \right\|_2^2 \left( \left\| \frac{K_{h_{j,OS}}^*}{f_\varepsilon^*} \right\|_1 \right)^{-2} &\geq C \left( \frac{\log(n)}{n} \right)^{\frac{1}{2\beta_j+1+2r}} c_\varepsilon^{-2} C_\varepsilon^{-2} \int_{\mathbb{R}} |K^*(u)|^2 |u|^{2r} du \left( \int_{\mathbb{R}} |K^*(v)| (1+|v|)^r dv \right)^{-2} \\ &\geq \frac{\log(n)}{n}, \end{aligned} \quad (52) $$

since $\frac{1}{2\beta_j + 1 + 2r} < 1$ for $\beta_j > 0, r > 0$ and Assumption 4.5 guarantees that all integrals finite.
Thus, $h_{j,OS} \in \mathcal{H}$. As a result, using Theorem 4.7, we obtain with probability greater than $1 - 4n^{-q}$, that

$$ |\hat{p}_{j,\hat{h}_j}(x) - p_j(x)| \le C_j \left( \frac{\log(n)}{n} \right)^{\frac{\beta_j}{2\beta_j+1+2r}}, $$

with constant $C_j$ depending on $\beta_j, \Lambda_j, \tilde{c}_{0,j}, K$ and $f_\varepsilon^*$. This concludes the proof of Lemma 6.6. $\square$

Let us now turn to the proof of Theorem 4.8. It is very similar to the proof of Theorem 3.4 except that one has to use Lemma 6.6 instead of Lemma 6.3. Consequently, we omit the proof.

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