# Asymptotic results of the randomly censored kernel-type expectile regression estimator for functional dependent data

Mustapha Mohammedi, Salim Bouzebda, Ali Laksaci

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Mustapha Mohammedi, Salim Bouzebda, Ali Laksaci. Asymptotic results of the randomly censored kernel-type expectile regression estimator for functional dependent data. Statistical Inference for Stochastic Processes, 2025, 28 (2), pp.10. 10.1007/s11203-025-09328-7. hal-05120336

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# Asymptotic results of the randomly censored kernel-type expectile regression estimator for functional dependent data

Mustapha Mohammedi<fnref n="1" /> · Salim Bouzebda<fnref n="2" /> · Ali Laksaci<fnref n="3" />

Received: 28 March 2025 / Accepted: 4 June 2025 / Published online: 18 June 2025
© The Author(s) 2025

## Abstract

This study examines the intricate process of estimating nonparametrically in expectile regression models for functional time series data that exhibit strong mixing properties within the context of a random right-censoring model. Specifically, we establish the almost complete consistency and asymptotic normality of the kernel-based expectile regression estimator. Notably, these results are derived in an asymptotic setting and are applicable under reasonably broad assumptions about the underlying model. Furthermore, we explore the practical implications of our theoretical findings in analyzing financial time series. To evaluate the performance of the proposed estimator on finite samples, we conducted comprehensive Monte Carlo simulations. These simulations provide a quantitative assessment of the estimator's accuracy and efficiency under various scenarios, allowing for a thorough understanding of its practical utility.

**Keywords** Almost complete (a.co.) convergence · Asymptotic normality · Censored data · Conditional expectile · Financial time series · Functional Data Analysis (FDA) · Kernel method · Small ball probability · Strong dependence

**Mathematics Subject Classification** Primary: 62G08 · 62G10 · 62G35 · 62G07 · 62G32 · 62G30; Secondary: 62H12.

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

<footnote>Authors supported by relevant institutions.</footnote>

<footnote>Salim Bouzebda
salim.bouzebda@utc.fr</footnote>

<footnote>Mustapha Mohammedi
mustapha.mohammedi@univ-mosta.dz</footnote>

<footnote>Ali Laksaci
alilak@yahoo.fr</footnote>

<footnote n="1">¹ Université Abdelhamid Ibn Badis de Mostaganem, LSPS - Université de Sidi Bel Abbès, Sidi Bel Abbès, Algeria</footnote>

<footnote n="2">² Université de technologie de Compiégne, LMAC, Compiégne, France</footnote>

<footnote n="3">³ Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia</footnote>
# 1 Introduction

In a variety of mathematics and statistics-related disciplines, risk measures are regarded as crucial instruments. Insurance, finance, and economics must all evaluate their prospective positions for uncertain outcomes using these metrics. Value-at-Risk (VaR) and Expected Shortfall (ES) are the most frequently employed financial risk analysis instruments. However, these statistical models have some drawbacks, including non-subadditivity and insensitivity to the magnitude of extreme losses (for definitions of these properties, see, for example, Bellini et al. (2019)). In this paper, we examine an alternative method based on the least asymmetrically weighted squares estimation, which is one of the fundamental tools in statistical applications and is derived from the econometrics literature. This method frequently employs the Newey and Powell (1987) concept of expectiles, the least squares analogue to conventional quantiles. In contrast to the quantiles of a random variable, they are founded on a quadratic loss function, just like the expectation. According to Newey and Powell (1987), while quantile regression offers the benefits of robustness against heterogeneous and heavy-tailed errors, it is characterised by its non-differentiability, limited effectiveness in handling Gaussian-type errors, and computationally demanding nature. The expectile model is gaining popularity in the financial literature; for example, see Pratesi et al. (2009); Waltrup et al. (2015); Bellini and Bernardino (2017); Farooq and Steinwart (2019). Since it is the only elicitable coherent risk measure, we may also refer to Bellini et al. (2019) and the references therein. One can refer to Gu and Zou (2016) and Zhao et al. (2018) for information regarding the use of expectile regression in heteroscedasticity analysis. We refer to the recent papers by Daouia et al. (2018) for additional justification on the use of the expectile model. For an overview of the application of expectile curves in regression analysis, see Kneib (2013) and the extensive discussions of that paper, in particular Eilers (2013) for an evaluation of expectiles and Koenker (2013) for a critical perspective. Quantiles and expectiles are extensions of the median and mean that comprise information about the full distribution of a random variable. Expectiles are excellent alternatives to quantiles in different aspects for pertinent applications. Expectiles are more sensitive than quantiles to the magnitude of infrequent catastrophic losses, and they depend on both the tail realizations of the predictor and their probability, whereas quantiles depend only on the frequency of tail realizations, see Kuan et al. (2009). This heightened sensitivity of expectiles to tail behavior enables more prudent and responsive risk management. Observe that the quantiles are not always adequate and can be criticized for being somewhat challenging to compute since the corresponding loss function is not continuously differentiable. The key advantage of the expectile over the quantile is its efficiency and computing experience, although it does not have the same direct interpretation in terms of relative frequency as the quantile, see Daouia et al. (2019). Although they are constructed differently, quantiles and expectiles share similar characteristics. As noted in Jones (1994), expectiles and *M*-quantiles relate to the mean and *M*-location estimators, respectively, in the same way that traditional quantiles relate to the median. Under the assumption of weighted symmetry, Abdous and Rémillard (1995) demonstrated that the quantiles and expectiles of the same distribution coincide and that inference on expectiles is much simpler than inference on quantiles.

Zhang (1994) presented the nonparametric estimation of expectile regression and demonstrated its consistency and asymptotic normality in a finite-dimensional framework. Regression models with a real-valued response variable and a smooth explanatory variable that varies unexpectedly between repeated observations or measurements have grown in popularity in recent years. This type of data, known as functional data, is utilized in numer-
ous disciplines, including climatology (the hourly concentration of pollutants), medicine (the knee angle of infants as a function of time), economics, and linguistics. Functional time series are encountered frequently in practice, such as when a protracted, continuous-time process is divided into smaller natural units, such as days. In this instance, each intraday curve is a functional random variable. Despite its significance, expectile regression remains relatively unexplored and underdeveloped in comparison to both competitive regressions (conditional mean and quantile), and our goal is to close this disparity. Mohammedi et al. (2021) provided the first results in this direction. They examined the nonparametric expectile regression for a functional predictor and a scalar response and obtained the asymptotic properties of the kernel expectile regression estimator in an iid setting. The primary objective of this study is to extend previous work to serially dependent observations to handle the case when the return or loss is linked to functional time series-valued exogenous covariates, thereby avoiding the “curse of infinite dimensions”. As described in Geenens (2011), the nonparametric functional time series analysis relies on the use of semi-metrics as proximity measures in infinite-dimensional functional space, which is frequently presented as a technical instrument for dimension reduction. This is one of the primary reasons for the evolution of the nonparametric method in functional statistics. Additional justifications for the significance of this topic of nonparametric functional statistics can be found in the preceding monograph Ferraty and Vieu (2006) or the exhaustive survey paper Ling and Vieu (2018); Bouzebda (2024c). In general, it is well-known that the statistical study of functionally dependent data stems from the monograph Bosq (2000) and that the literature has flourished since then. Particularly, the first study on Nonparametric Functional Time Series Analysis (NFTSA) was conducted by Ferraty et al. (2002), which established the almost-complete consistency of the kernel estimator of the regression operator. Masry (2005) examined regression estimation employing a kernel-type estimator in a functional setting with blending samples and established asymptotic normality. Refer to Bouzebda and Chaouch (2022) for a discussion of the asymptotic normality of the regression operator when the functional time series data has an ergodic structure. Ling and Liu (2017) investigated the large-sample properties of the kernel estimator for a non-parametric regression operator when the response variable is randomly censored based on the ergodicity assumption. Recent research has examined the properties of kernel estimators in a functional regression model when both the response and covariate are functional, see Ling et al. (2020). The authors demonstrated the examined estimator's almost complete pointwise and uniform convergence. Ling et al. (2019) examined the *k*-nearest neighbors (*k*NN) estimator of a nonparametric regression model for functional time series data and determined its uniform nearly complete convergence rate. Chowdhury and Chaudhuri (2020) examined a nonparametric regression configuration with a focus on an important parameter associated with the conditional distribution. In this configuration, the authors determined the optimal convergence rate for the kernel estimate of the parameter. Other NFTSA studies employing alternative nonparametric models include Ezzahrioui and Ould-Saïd (2010) and Bouzebda et al. (2016) (for the conditional mode), Quintela-del Río (2008) (for the hazard function), Chen et al. (2019) and Bouzebda and Chaouch (2022) (for the *M*-regression), and Ling et al. (2022) (for the single index model).

Functional time series analyses by quantile regression are active and relevant research disciplines that are closely related to the expectile regression model. Ezzahrioui and Ould-Saïd (2008), for instance, have established the asymptotic normality for a kernel estimator of the conditional quantile derived by inverting the double kernel estimator of the cumulative distribution function. Laksaci et al. (2011) have constructed an alternative estimator of the quantile regression using the $L_1$ method and determined its asymptotic properties under the strong mixing assumption. For the nonparametric estimation of the conditional quantile for
ergodic functional data, we cite Bellini and Bernardino (2017). In the functional statistics literature, additional quantile regression models have been proposed. We cite Ding et al. (2018) (for semi-functional partial linear model), Sang and Cao (2020) (for single index model), and Zhang et al. (2021) (for additive model) as examples. Ferraty and Vieu (2006) examined nonparametric models within the context of regression estimation. Additionally, we may cite Bosq (2000); Horváth and Kokoszka (2012). Kokoszka and Reimherr (2017) offered an excellent combination of foundational material, approachable theory, and practical examples. For recent developments and trends in FDA, refer to the following survey articles and journal special issues: Cuevas (2014); Goia and Vieu (2016); Ling and Vieu (2018); Aneiros et al. (2019), and the recent contributions Bouzebda and Nezzal (2022); Bouzebda et al. (2023b); Bouzebda and Nezzal (2024b); Bouzebda and Nemouchi (2023); Bouzebda et al. (2023a); Bouzebda (2024a); Soukarieh and Bouzebda (2024); Bouzebda (2024d).

The majority of the available literature is devoted to the case where the variable of interest, say $Y$, is completely observed. This is not the case in many interesting applications, including survival analysis, where censoring prevents the direct application of “classical” parametric methods because instead of observing $Y$, one only observes the minimum of $Y$ and a censoring variable. Survival data in clinical trials or failure time data in reliability studies, for example, are often subject to such censorship. To be more specific, many statistical experiments result in incomplete samples, even under well-controlled conditions. For example, clinical data for surviving most types of disease is usually censored by other competing risks to life that result in death. The primary objective of this paper is to develop an estimator for expectile regression under censored data and establish its almost complete convergence and asymptotic normality, assuming a strong mixing condition. This problem has remained unresolved until now, providing the principal motivation for our research. In the present work, we extend our previous results from Almanjahie et al. (2022b) to the setting of randomly censored data. However, as will become clear, this extension involves more than a straightforward combination of existing ideas and results. In particular, sophisticated mathematical derivations are necessary to handle the presence of the Kaplan-Meier estimator in the limiting behavior of the proposed estimators. In addition, when the censoring time satisfies $\mathbb{P}(C = \infty) = 1$, we recover the uncensored case studied in our previous work.

The layout of the article is as follows. In the subsequent section, we introduce the kernel estimator for expectile regression in the setting of the strong mixing censored data. Section 3 presents our main findings, including the almost complete convergence and asymptotic distribution properties of the expectile regression kernel estimator. To illustrate the usefulness of our proposed methods, we conducted a simulation study in Section 4. In Section 5, real financial time series data is discussed. Finally, in Section 6, we provide concluding remarks and discussions. The proofs of the main results are presented in Section 7.

## 2 Model and estimator

We start by giving some notation and definitions that are needed for the forthcoming sections. Let’s recall the definition of the strong mixing property. For this purpose, we introduce the following notation. Let $\mathcal{F}_i^k(Z)$ be the $\sigma$-algebra generated by $\{Z_j : i \le j \le k\}$. The following definition of the strong mixing coefficient is due to Rosenblatt (1956). For more details, one can refer to Bradley (2007).
**Definition 2.1** Let $Z = \{Z_i, i = 1, 2, \dots\}$ be a strictly stationary sequence of random variables. Given a positive integer $n$, set

$$ \alpha(n) = \sup \left\{ \left| \mathbb{P}(A \cap B) - \mathbb{P}(A)\mathbb{P}(B) \right| : A \in \mathcal{F}_1^k(Z) \text{ and } B \in \mathcal{F}_{k+n}^\infty(Z), k \in \mathbb{N}^* \right\}. $$

The sequence $Z$ is said to be $\alpha$-mixing (strong mixing) if the mixing coefficient $\alpha(n) \to 0$ as $n \to \infty$.

The $\alpha$-mixing is the weakest of the most commonly used mixing conditions and has many practical applications, especially in the fields of economics and finance. For example, there exist several time series models satisfying the $\alpha$-mixing condition, such as ARCH, ARMA, GARMA, and stochastic volatility models. Let us consider a triple $(T, C, X)$ of random variables defined in $\mathbb{R}^+ \times \mathbb{R}^+ \times \mathcal{F}$, where $T$ is the variable of interest (for instance, the lifetime variable), with a common unknown absolutely continuous distribution function $F, C$ is a censoring variable and $X$ is a functional covariate taking values in an infinite-dimensional space $\mathcal{F}$ equipped a semi-metric $d(\cdot, \cdot)$ (A semi-metric (sometimes called pseudo-metric) $d(\cdot, \cdot)$ is a metric which allows $d(x_1, x_2) = 0$ for some $x_1 \neq x_2$), defining a topology to measure the proximity between two elements of $\mathcal{F}$ and which is disconnected from the definition of $X$ to avoid measurability problems. Let $G(\cdot)$ be the distribution function of $C$. In the right censored model, the pair of random variables $(T, C)$ is not directly observed, and the available information is given by $Y = \min(T, C)$ and $\delta = \mathbb{1}_{\{T \le C\}}$, where $\mathbb{1}_A$ stands for the indicator function of the set $A$. We assume that a sample $D_n = \{(Y_i, \delta_i, X_i), i = 1, \dots, n\}$ is at our disposal, and we suppose throughout this paper that $(X_i, T_i)_{i=1,\dots,n}$ is a $\alpha$-mixing sequence assumed to be independent of the sequence of random censoring $(C_i)_{i=1,\dots,n}$. Let $x$ be a fixed function in $\mathcal{F}$, $N_x$ is a fixed neighbourhood of $x$, and the closed ball centered at $x$ and of radius $\beta$ is denoted by

$$ B(x, \beta) = \{y \in \mathcal{F} \text{ such that } d(y, x) \le \beta\}. $$

Koenker and Bassett (1978) elaborated an absolute error loss minimization framework to define quantiles. Later, Newey and Powell (1987) substituted the $L_1$-loss function with the $L_2$-loss to define the population expectile of order $p \in (0, 1)$; see also Efron (1991); Abdous and Rémillard (1995); Mohammedi et al. (2021, 2020); Bouzebda et al. (2022); Almanjahie et al. (2022a, b, 2024) for further references. Breckling and Chambers (1988) considered regression based on more general asymmetric $M$-estimators. The objective of this paper is to estimate the $p^{th}$ conditional expectile of $Y$ given $X = x$, denoted by $\theta(p; x)$, defined by

$$ \theta(p; x) = \arg \min_{z \in \mathbb{R}} \mathbb{E} \left[ p(Y - z)^2 \mathbb{1}_{\{(Y-z)>0\}} + (1-p)(Y-z)^2 \mathbb{1}_{\{(Y-z)\le 0\}} \mid X = x \right], \quad (2.1) $$

where $A$ being the indicator function of the set $A$. It is worth noticing that (2.1) generalizes the conditional expectation of $Y$ given $X = x$, which coincides with $\theta(p; x)$ when specifically $p = 1/2$, such as the conditional quantile generalizes the conditional median. On the other hand, (2.1) is similar to the conditional $p$-quantile of $Y$ given $X = x$, which can be obtained by replacing $(Y - z)^2$ by $|Y - z|$ in (2.1). For more recent references on the quantile regression, we refer to Lv et al. (2019); Xu et al. (2019); Shen and Liang (2018). This motivates the name the conditional $p$-expectile. Recall that for any measurable function $\psi$, we define the regression function

$$ m_{\psi}(\mathbf{x}) = \mathbb{E}[\psi(T) | X = x]. $$
When the response $T$ is subject to right-censoring we can still estimate $m_\psi$ by applying the inverse probability of censoring weighting (IPCW) idea. Introduce the mapping $\Phi_\psi : \mathbb{R}^2 \to \mathbb{R}$ by

$$ \Phi_{\psi}(t, c) = \frac{\mathbb{1}_{\{t \le c\}} \psi(t \wedge c)}{1 - \mathbf{G}(t \wedge c)}. $$

The quantity

$$ \Phi_{\psi}(T_i, C_i) = \frac{\delta_i \psi(Y_i)}{1 - \mathbf{G}(Y_i)} \quad (\delta_i = \mathbb{1}_{\{T_i \le C_i\}}, Y_i = T_i \wedge C_i) $$

is directly observable for every $i = 1, \dots, n$. Then, by assuming $C$ and $(T, X)$ are independent; for every $x$,

$$ \begin{aligned} m_{\Phi_{\psi}}(\mathbf{x}) &:= \mathbb{E}[\Phi_{\psi}(T, C) | X = x] = \mathbb{E}\left[\frac{\mathbb{1}_{\{T \le C\}} \psi(Y)}{1 - \mathbf{G}(Y)} \middle| X = x\right] \\ &= \mathbb{E}\left[\frac{\psi(T)}{1 - \mathbf{G}(T)} \underbrace{\mathbb{E}[\mathbb{1}_{\{T \le C\}} | X, T]}_{= \mathbb{P}(C \ge T | X, T) = 1 - \mathbf{G}(T)} \middle| X = x\right] \\ &= \mathbb{E}[\psi(T) | X = x] = m_{\psi}(x). \end{aligned} $$

Hence the conditional expectation of the observed IPCW transform $\Phi_\psi(Y, C)$ coincides with the original target $m_\psi(x)$. This identity is the foundation of IPCW estimators for regression with right-censored responses. By a simple manipulation, we show that $\theta(p; x)$ is the unique solution with respect to $z$ of, i.e.,

$$ \zeta(p; x) = \mathcal{E}(z; x) := \frac{\mathcal{E}_1(z; x)}{\mathcal{E}_2(z; x)}, \qquad (2.2) $$

where

$$ \left\{ \begin{aligned} \zeta(p; x) &= \frac{p}{1-p}, \\ \mathcal{E}_1(z; x) &= -\mathbb{E}[\delta(\bar{\mathbf{G}}(Y))^{-1}\psi_1(Y-z) | X=x], \\ \mathcal{E}_2(z; x) &= \mathbb{E}[\delta(\bar{\mathbf{G}}(Y))^{-1}\psi_2(Y-z) | X=x], \end{aligned} \right. $$

and

$$ \left\{ \begin{aligned} \psi_1(\sigma) &= \sigma \mathbb{1}_{\{\sigma \le 0\}}, \\ \psi_2(\sigma) &= \sigma \mathbb{1}_{\{\sigma > 0\}}, \end{aligned} \right. $$

where $\bar{\mathbf{G}}(\cdot) = 1 - \mathbf{G}(\cdot)$. Thus, the monotonicity of the function $\mathcal{E}(\cdot; x)$ allows to write that

$$ \theta(p; x) = \inf \left\{ z \in \mathbb{R} : \mathcal{E}(z; x) \ge \frac{p}{1-p} \right\}. $$

Now, the natural "pseudo-estimator" kernel estimator of $\theta(p; x)$ is the solution with respect to $z$ of

$$ \frac{p}{1-p} = \hat{\mathcal{E}}_{n;h}(z; x), $$

where $\hat{\mathcal{E}}_{n;h}(z; x)$ is the kernel estimator of $\mathcal{E}(z; x)$. More precisely, for kernel function $\mathcal{K}(\cdot)$ and smoothing parameter $h_n := h$, the kernel estimator of the function $\mathcal{E}(\cdot; x)$ is given by

$$ \hat{\mathcal{E}}_{n;h}(z; x) = \frac{-\sum_{i=1}^{n} \tilde{W}_{ni}(x) \psi_1(Y_i - z)}{\sum_{i=1}^{n} \tilde{W}_{ni}(x) \psi_2(Y_i - z)} = \frac{\hat{\mathcal{E}}_{n;h;1}(z; x)}{\hat{\mathcal{E}}_{n;h;2}(z; x)}, \quad \text{for } z \in \mathbb{R}, $$
where

$$
\tilde{W}_{ni}(x) = \frac{\delta_i (\overline{\mathbf{G}}(Y_i))^{-1} \mathcal{K}_i(x)}{\sum_{i=1}^{n} \mathcal{K}_i(x)} \quad \text{and} \quad \mathcal{K}_i(x) = \mathcal{K}(h^{-1} d(x, X_i)),
$$

and

$$
\tilde{\mathcal{E}}_{n;h;1}(z; x) = - \frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} \sum_{i=1}^{n} \delta_i (\overline{\mathbf{G}}(Y_i))^{-1} \mathcal{K}_i(x) \psi_1(Y_i - z),
$$

$$
\tilde{\mathcal{E}}_{n;h;2}(z; x) = \frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} \sum_{i=1}^{n} \delta_i (\overline{\mathbf{G}}(Y_i))^{-1} \mathcal{K}_i(x) \psi_2(Y_i - z).
$$

In practice, $\mathbf{G}(\cdot)$ is unknown; therefore, to get a feasible estimator, we can replace $\overline{\mathbf{G}}(\cdot)$ by,
for instance, its Kaplan and Meier (1958) estimator $\overline{\mathbf{G}}_n(\cdot)$ that is given by

$$
\overline{\mathbf{G}}_n(t) = \begin{cases} \prod_{i=1}^{n} \left(1 - \frac{1-\delta_{(i)}}{n-i+1}\right)^{\frac{1}{n}} \{y_{(i)} \le t\} & \text{if } t < Y_{(n)}, \\ 0 & \text{otherwise,} \end{cases}
$$

where $Y_{(1)} < Y_{(2)} < \dots < Y_{(n)}$ are the order statistics of $(Y_i)_{1 \le i \le n}$ and $\delta_{(i)}$ is the concomitant of $Y_{(i)}$. Therefore an estimator of $\mathcal{E}(\cdot; x)$ can be defined as

$$
\hat{\mathcal{E}}_{n;h}(z; x) = \frac{-\sum_{i=1}^{n} \hat{W}_{ni}(x) \psi_1(Y_i - z)}{\sum_{i=1}^{n} \hat{W}_{ni}(x) \psi_2(Y_i - z)} = \frac{\hat{\mathcal{E}}_{n;h;1}(z; x)}{\hat{\mathcal{E}}_{n;h;2}(z; x)}, \quad \text{for } z \in \mathbb{R},
$$

where

$$
\hat{W}_{ni}(x) = \frac{\delta_i (\overline{\mathbf{G}}_n(Y_i))^{-1} \mathcal{K}_i(x)}{\sum_{i=1}^{n} \mathcal{K}_i(x)},
$$

and

$$
\hat{\mathcal{E}}_{n;h;1}(z; x) = - \frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} \sum_{i=1}^{n} \delta_i (\overline{\mathbf{G}}_n(Y_i))^{-1} \mathcal{K}_i(x) \psi_1(Y_i - z),
$$

$$
\hat{\mathcal{E}}_{n;h;2}(z; x) = \frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} \sum_{i=1}^{n} \delta_i (\overline{\mathbf{G}}_n(Y_i))^{-1} \mathcal{K}_i(x) \psi_2(Y_i - z).
$$

Thus, the kernel estimator of the conditional expectile $\theta(p; x)$, denoted by $\hat{\theta}_{n;h}(p; x)$, is
explicitly defined by

$$
\hat{\theta}_{n;h}(p; x) = \inf \left\{ z \in \mathbb{R} : \hat{\mathcal{E}}_{n;h}(z; x) \geq \frac{p}{1-p} \right\}. \quad (2.3)
$$

Notice that

$$
\begin{align*}
\hat{\mathcal{E}}_{n;h}(z; x) - \mathcal{E}(z; x) &= \frac{\hat{\mathcal{E}}_{n;h;1}(z; x)}{\hat{\mathcal{E}}_{n;h;2}(z; x)} - \frac{\mathcal{E}_1(z; x)}{\mathcal{E}_2(z; x)} \\
&= \frac{1}{\hat{\mathcal{E}}_{n;h;2}(z; x)} \left[ \begin{aligned}[t]
& (\tilde{\mathcal{E}}_{n;h;1}(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;1}(z; x)]) \\
& + (\hat{\mathcal{E}}_{n;h;1}(z; x) - \tilde{\mathcal{E}}_{n;h;1}(z; x)) + (\mathbb{E}[\tilde{\mathcal{E}}_{n;h;1}(z; x)] - \mathcal{E}_1(z; x))
\end{aligned} \right]
\end{align*}
$$
$$ + \frac{\mathcal{E}(z; x)}{\hat{\mathcal{E}}_{n;h;2}(z; x)} \left[ \begin{aligned} & (\mathbb{E}[\tilde{\mathcal{E}}_{n;h;2}(z; x)] - \tilde{\mathcal{E}}_{n;h;2}(z; x)) \mathcal{E}_1(z; x) \\ & + (\tilde{\mathcal{E}}_{n;h;2}(z; x) - \hat{\mathcal{E}}_{n;h;2}(z; x)) \mathcal{E}_1(z; x) \\ & + (\mathcal{E}_2(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;2}(z; x)]) \mathcal{E}_1(z; x) \end{aligned} \right]. \quad (2.4) $$

## 2.1 Assumptions and notation

The assumptions underlying our results are presented below for convenient reference.

(H0) The upper endpoint for any df W is defined as follows

$$ \tau_W := \sup\{t, W(t) < 1\}. $$

We assume that $\tau_F < \infty$ and $\bar{\mathbf{G}}(\tau_F) > 0$, which implies that $\tau_F < \tau_G$ and that $(C_i)_{1 \le i \le n}$ and $(X_i, T_i)_{1 \le i \le n}$ are independent;

(CO1) $\forall \varrho > 0$, $\mathbb{P}(X \in B(x, \varrho)) = \phi_x(\varrho) > 0$, where $B(x, \varrho)$ denotes the ball of the center $x$ and radius $\varrho > 0$. Moreover,

$$ \lim_{n \to \infty} h_n = 0 \quad \text{and} \quad \lim_{n \to \infty} \phi_x(h_n) = 0; $$

(CO2) The function $\mathcal{E}(\cdot; x)$ is differentiable in $\mathbb{R}$ and satisfies the following Lipschitz's condition: $\exists \delta > 0, \forall z \in [\theta(p; x) - \delta, \theta(p; x) + \delta]$, $\forall x_1, x_2 \in \mathcal{F}$,

$$ |\mathcal{E}_i(z; x_1) - \mathcal{E}_i(z; x_2)| \le C_i d^{k_i}(x_1, x_2), \quad \text{for } k_i > 0, i \in \{1, 2\}. $$

Furthermore, we assume that $\forall x \in \mathcal{F}, \mathcal{E}'_1(\theta(p; x); x)$ and $\mathcal{E}'_2(\theta(p; x); x)$ are not identical to zero, with $\mathcal{E}'_i(z; x)$ denoting the derivative of the function $\mathcal{E}_i(z; x)$ with respect to $z$;

(CO3) There exists $m' \ge 2$, $L_i^x(u) = \mathbb{E}[|\psi_i(Y-u)|^{m'} | X=x] \le C_3 < \infty$, almost surely, for $i \in \{1, 2\}$, where $\psi_i(\cdot)$ is defined in (2.2);

(CO4) The kernel $\mathcal{K}(\cdot)$ is a measurable function that is supported by $[0, 1]$ and satisfies:

$$ 0 < C_4 1_{[0,1]} < \mathcal{K}(t) < C_5 1_{[0,1]} < \infty; $$

(CO5) The sequence $(X_i, Y_i)_{i \in \mathbb{N}}$ satisfies: $\exists a > 0, \exists C_6 > 0 : \forall n \in \mathbb{N}, \alpha(n) \le C_6 n^{-a}$ and

$$ \left\{ \begin{array}{ll} \forall i \neq j, & \mathbb{E}[Y_i Y_j | X_i, X_j] \le C_7 < \infty, \quad \text{a.s.} \\ \mathbb{P}((X_i, X_j) \in B(x, r) \times B(x, r)) = \varphi_x(r) > 0; & \end{array} \right. $$

(CO6) There exists $0 < \delta < 2 \frac{a-2}{a+1}$, such that

$$ C_8 n^{\frac{2(2-a)}{(a+1)} + \delta} \le \chi_x(h) \le C_9 n^{\frac{2a}{(2-a)m}}, $$

where

$$ \chi_x(h) = \max(\phi_x^2(h), \varphi_x(h)) \quad \text{and} \quad a > \frac{1 + 4m' + \sqrt{24m' + 1}}{m' - 1}. $$
(NA1) The function $\mathcal{K}(\cdot)$ satisfies (CO4) and is differentiable on $(0, 1)$ with derivative $\mathcal{K}'(\cdot)$ such that

$$0 < \alpha_j(x) = \mathcal{K}^j(1) - \int_0^1 (\mathcal{K}^j)'(s) \beta_x(s) ds < \infty, \text{ for } j = 1, 2;$$

(NA2) The concentration property given in (CO1) holds and there exists a function $\beta_x(\cdot)$ such that

$$\forall s \in [0, 1], \lim_{t \to 0} \frac{\phi_x(st)}{\phi_x(t)} = \beta_x(s);$$

(NA3) The condition (CO2) holds for $i = 1, 2$ and $m' = 2$, the functions

$$L_i^x(u) = \mathbb{E}(\psi_i^2(Y - u) | X = x),$$

are differentiable with respect to $u$ in $\mathbb{R}$ and satisfies the following Lipschitz's condition: $\exists a > 0, \forall t \in [\theta(p; x) - a, \theta(p; x) + a], \forall x_1, x_2 \in \mathcal{F},$

$$|L^{x_1}(u) - L^{x_2}(u)| \le C_{10}d^r(x_1, x_2), \text{ for } r > 0;$$

(NA4) We assume that $n\phi_x(h) \to \infty$, there exists $\delta > 0$ such that $\chi_x(h) \le C_{11}n^{-\frac{1}{a(m'-2)/m'}}\phi_x^{\frac{a}{m'}}(h)$ and for $k > \min(k_1, k_2)$, $nh^{2k}\phi_x(h) \to 0$ as $n \to \infty$.

## 2.2 Comments on the assumptions

The considered conditions cover the three structural axes of this study. Indeed, (H0) is very standard in nonparametric censoring. The first part of the assumptions (H0) is used to obtain the consistency of the Kaplan Meier estimator, and when $\tau_F > \tau_G$ all values in the interval $[\tau_G, \tau_F]$ are censored and therefore not observed. For the second part of (H0), the independence between C and (X, T) is usual whenever the censoring is independent of the characteristics of the individuals under study, and therefore which allows us to conclude the following equality:

$$\mathbb{E}\left[\frac{\delta_1 Y_1}{\hat{G}(Y_1)} | X_1 = x\right] = \mathbb{E}[T_1 | X_1 = x].$$

The functional nature of the data is explored by the concentration property of the explanatory variable in small balls defined in (CO1); there is no probability measure such as the Lebesgue one in the multivariate case, and one has to make a similar condition without the concept of probability density. This assumption replaces the condition of strict positivity of the density of explanatory variables usually assumed in a finite-dimensional setting, see Masry (2005), who in turn was inspired by Gasser et al. (1998) in the non-parametric density estimation under functional observations. Masry (2005) explains that if $\mathcal{F} = \mathbb{R}$, then the condition coincides with the fundamental axioms of probability calculus, furthermore if $\mathcal{F}$ is an infinitely dimensional Hilbert space then $\phi(h)$ can decrease toward 0 by an exponential speed as $n \to \infty$. While the nonparametric property of the model is characterized through the Hölderian condition (CO2), used to investigate the bias term of the estimator $\hat{\mathcal{E}}_{n;h}$. The assumption (CO5) is a standard assumption on the decay of a strong mixing coefficient; the first and second conditions of (CO5) concern respectively the global and local dependence of functional observations. The technical conditions (CO3), (CO4), and (CO6) are imposed for the brevity of the proof and to specify the convergence rate of the estimator. Condition
(NA1) is related to the choice of the kernel $\mathcal{K}(\cdot)$, which is very usual in nonparametric functional estimation. Notice that Parzen's symmetric kernel is not adequate in this context since the random process $D_t = d(x, X_t)$ is positive, therefore, we consider $\mathcal{K}(\cdot)$ with support $[0, 1]$. This is a natural generalization of the assumption usually made on the kernel in the multivariate case, where $\mathcal{K}(\cdot)$ is supposed to be a spherically symmetric density function. The assumptions $\mathcal{K}(1) > 0$ and $\mathcal{K}'(t) < 0$ guarantee that $\alpha_j(x) > 0$ for all limit functions $\beta_x$. In the case of non-smooth processes $\beta_x$ may be equal to the Dirac $\delta$-function at 1, the condition $\mathcal{K}(1) > 0$ is needed to define the moments $\alpha_j$ which are, in this case, determined by the value $\mathcal{K}(1)$. The assumption (NA3) is a Hölder-type assumption that requires a certain smoothness of the regression operator and $L^x(\cdot)$. The assumption (NA2) permits to give the variance term explicitly in the asymptotic normality, and approximately shows that the small ball probability can be written approximately as the product of two independent functions, refer to Mayer-Wolf and Zeitouni (1993) for the diffusion process, Bogachev (1998) for a Gaussian measure, Li and Shao (2001) for a general Gaussian process and has employed these assumptions Masry (2005) for strongly mixing processes. For example, the function $\beta_x(\cdot)$ appears in all asymptotic behaviors, and the choice of this function can be found in Ferraty et al. (2007). The function $\phi(\cdot)$ can be expressed as $\phi(\epsilon) = \epsilon^\delta \exp(-C/\epsilon^a)$ with $\delta \ge 0$ and $a \ge 0$, and it corresponds to the Ornstein-Uhlenbeck and general diffusion processes (for such processes, $a = 2$ and $\delta = 0$) and the fractal processes (for such processes, $\delta > 0$ and $a = 0$). This class of processes also satisfies condition (NA2). For other examples, we refer to Bouzebda and Nemouchi (2023); Bouzebda (2024a). The assumptions (NA4) on $\phi_x(h)$ and $\chi_x(h)$ are imposed for the sake of the brevity of our proofs. Let $(u_n)$ for $n \in \mathbb{N}$, be a sequence of real r.v.'s. We say that $(u_n)$ converges almost-completely (a.co.) toward zero if, and only if, for all

$$ \epsilon > 0, \sum_{n=1}^{\infty} \mathbb{P}(|u_n| > \epsilon) < \infty. $$

Moreover, we say that the rate of the almost-complete convergence of $(u_n)$ toward zero is of order $v_n$ (with $v_n \to 0$) and we write $u_n = O_{a.co.}(v_n)$ if, and only if, there exists $\epsilon > 0$ such that

$$ \sum_{n=1}^{\infty} \mathbb{P}(|u_n| > \epsilon v_n) < \infty. $$

This kind of convergence implies both the almost-sure convergence and the convergence in probability.

# 3 Main results

## 3.1 Consistency

Our first result in this section is summarized as follows.
**Proposition 3.1** Under the assumptions (CO1)-(CO6), we have, for all $z \in \mathbb{R}$ and $x \in \mathcal{F}$, as $n \to \infty$,

$$|\hat{\mathcal{E}}_{n;h}(z; x) - \mathcal{E}(z; x)| = O(h^{k_1}) + O(h^{k_2}) + O_{a.co.} \left( \sqrt{\frac{\log n}{n\phi_x(h)}} \right). \quad (3.1)$$

The following theorem ensures the consistency of $\hat{\theta}_{n;h}(p; x)$ toward $\theta(p; x)$ as the sample size goes to infinity.

**Theorem 3.2** Under the assumptions of Proposition 3.1. Then, as $n \to \infty$, we have

$$|\hat{\theta}_{n;h}(p; x) - \theta(p; x)| = O_{a.co.} \left( h^{\min\{k_1, k_2\}} + \sqrt{\frac{\log n}{n\phi_x(h)}} \right). \quad (3.2)$$

To prove Theorem 3.2, we shall make use of the following lemmas.

**Lemma 3.3** Under the conditions (CO1), (CO3)-(CO5), for all $z \in \mathbb{R}$ and $x \in \mathcal{F}$, we have, as $n \to \infty$,

$$|\hat{\mathcal{E}}_{n;h;1}(z; x) - \tilde{\mathcal{E}}_{n;h;1}(z; x)| = O_{a.s.} \left( \sqrt{\frac{\log \log(n)}{n}} \right), \quad (3.3)$$

and

$$|\tilde{\mathcal{E}}_{n;h;2}(z; x) - \hat{\mathcal{E}}_{n;h;2}(z; x)| = O_{a.s.} \left( \sqrt{\frac{\log \log(n)}{n}} \right). \quad (3.4)$$

**Lemma 3.4** Assume that the conditions (CO1)-(CO4) are fulfilled. We have, for all $z \in \mathbb{R}$ and $x \in \mathcal{F}$, as $n \to \infty$,

$$|\mathbb{E}[\tilde{\mathcal{E}}_{n;h;1}(z; x)] - \mathcal{E}_1(z; x)| = O(h^{k_1}), \quad (3.5)$$

and

$$|\mathcal{E}_2(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;2}(z; x)]| = O(h^{k_2}). \quad (3.6)$$

**Lemma 3.5** Under the conditions (CO1), (CO3)-(CO6), we have, for all $z \in \mathbb{R}$ and $x \in \mathcal{F}$, as $n \to \infty$,

$$|\tilde{\mathcal{E}}_{n;h;1}(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;1}(z; x)]| = O_{a.co.} \left( \sqrt{\frac{\log n}{n\phi_x(h)}} \right), \quad (3.7)$$

and

$$|\mathbb{E}[\tilde{\mathcal{E}}_{n;h;2}(z; x)] - \tilde{\mathcal{E}}_{n;h;2}(z; x)| = O_{a.co.} \left( \sqrt{\frac{\log n}{n\phi_x(h)}} \right). \quad (3.8)$$

## The classical regression case

It is well documented that classical regression can be treated as a particular case of expectile regression. By simple manipulation, we prove that

$$r(x) := \theta(0.5; x) = \mathbb{E}[Y | X = x] \text{ and } \hat{r}(x) := \hat{\theta}_{n;h}(0.5; x).$$
For this case, the condition (CO2) is reduced to

$$ \forall x_1, x_2 \in \mathcal{F}, |r(x_1) - r(x_2)| \le C_1 d^{k_1}(x_1, x_2). \quad (3.9) $$

Based on this model, we get the following result.

**Corollary 3.6** Under conditions (CO1), (CO3)-(CO5) and if (3.9) holds, then we have, for $x \in \mathcal{F}$, as $n \to \infty$,

$$ |\hat{r}(x) - r(x)| = O_{a.co.} \left( h^{k_1} + \sqrt{\frac{\log n}{n\phi_x(h)}} \right). \quad a.co. $$

This corollary shows that the results of this paper extend those of Ferraty et al. (2002) on the regression operator.

## The conditional expected shortfall estimation through the expectile regression

We can say there is an important gain by using the expectile regression compared to the conditional quantile. Moreover, the conditional expected shortfall $C_x(\alpha)$ can be alternatively computed by using the expectile regression estimator. Noting that the one-to-one correspondence between the expectiles and the quantiles of a related distribution has been proved by Jones (1994). Also, it is proved by the same author that for each $\alpha$-conditional quantile $q_\alpha(x)$ there is a unique expectile. For a recent comparison between quantile and expectile regression and references, see Waltrup et al. (2015). In particular, the right-tail $C_x(\alpha)$, is defined by

$$ C_x(\alpha) = \mathbb{E}[Y | Y \ge q_\alpha(x), X = x], $$

and is expressed as follows, for all $\alpha \in (0, 1)$,

$$ \begin{aligned} C_x(\alpha) &= \Phi(\alpha, \theta(p; x)) \\ &= \left( \frac{1-p}{(1-2p)(1-\alpha)} \right) \theta(0.5; x) - \left( \left( \frac{\alpha(1-2p)+p}{(1-2p)(1-\alpha)} \right) \theta(p; x) \right), \quad (3.10) \end{aligned} $$

where $p \in (0, 1)$ is such that

$$ \theta(p; x) = q_{\alpha}(x), \quad (3.11) $$

with $q_\alpha(x)$ in (3.11) is the conditional quantile of $Y$ given $X = x$ of order $\alpha \in (0, 1)$. Therefore, the conditional expected shortfall is estimated by

$$ \widehat{C}_x(\alpha) = \Phi(\alpha, \widehat{\theta}_{n;h}(p; x)). $$

Now, by applying Theorem 3.2, we have the following corollary.

**Corollary 3.7** Assume that the conditions (CO1)-(CO5) are satisfied. Suppose that

$$ \frac{\partial \mathcal{E}(\theta(p; x); x)}{\partial z} > 0. $$

We have, for $x \in \mathcal{F}$, as $n \to \infty$,

$$ C_x(\alpha) - \widehat{C}_x(\alpha) \to 0, \quad a.co. $$
## 3.2 Asymptotic normality

Let us now state the following theorem, which gives the asymptotic normality of the estimator $\hat{\theta}_{n;h}(p; x)$ defined in (2.3). Below, we write $Z \stackrel{D}{=} N(\mu, \sigma^2)$ whenever the random variable $Z$ follows a normal law with expectation $\mu$ and variance $\sigma^2$, $\stackrel{D}{\to}$ denotes the convergence in distribution. The main result of this section may now be stated as follows.

**Theorem 3.8** Under the assumptions (CO3), (CO5) and (NA1)-(NA4), we have, as $n \to \infty$,

$$ \left( \frac{n\phi_x(h)}{\sigma_p^2(x)} \right)^{\frac{1}{2}} (\hat{\theta}_{n;h}(p; x) - \theta(p; x)) \stackrel{D}{\to} N(0, 1), \text{ for } x \in \mathcal{F} \text{ such that } \sigma_p^2(x) > 0, \quad (3.12) $$

where

$$ \sigma_p^2(x) = \frac{\alpha_2(x)\gamma_p(\theta(p; x); x)}{\alpha_1^2(x)\Sigma_p^2(\theta(p; x); x)}, \quad (3.13) $$

$$ \alpha_j(x) = \mathcal{K}^j(1) - \int_0^1 (\mathcal{K}^j)'(s)\beta_x(s)ds \text{ for } j = 1, 2, \quad (3.14) $$

and

$$ \gamma_p(u; x) = \left( \frac{p}{1-p} \right)^2 L_2^x(u) + L_1^x(u), $$

where

$$ L_2^x(u) = \mathbb{E}(\psi_2^2(Y_1 - u) | X = x), \quad L_1^x(u) = \mathbb{E}(\psi_1^2(Y_1 - u) | X = x), $$

and

$$ \Sigma_p(u; x) = \mathcal{E}'_1(u; x) - \left(\frac{p}{1-p}\right) \mathcal{E}'_2(u; x). $$

## Application to the confidence intervals

In addition to the above discussion that treats the pointwise case, we highlight the impact of our model in financial time series using confidence interval processing. Certainly, this approach is more informative than pointwise modeling. The latter is a direct application of the central limit theorem (CLT). It is based on the plug-in estimation of the variance term $\sigma_p^2(x)$. Precisely, we keep the notation of the Theorem 3.8 and we estimate the parameters $\alpha_1(x)$ and $\alpha_2(x)$ by

$$ \hat{\alpha}_1(x) = \frac{1}{n\phi_x(h)} \sum_{i=1}^{n} \mathcal{K}_i(x) \quad \text{and} \quad \hat{\alpha}_2(x) = \frac{1}{n\phi_x(h)} \sum_{i=1}^{n} \mathcal{K}_i^2(x), $$

and $\Sigma_p$ by

$$ \hat{\Sigma}_p(\hat{\theta}_h(p; x); x) = \hat{\mathcal{E}}'_1(\hat{\theta}_h(p; x); x) - \left(\frac{p}{1-p}\right) \hat{\mathcal{E}}'_2(\hat{\theta}_h(p; x); x). \quad (3.15) $$
Next, the estimator of $\gamma_p$ is given by

$$ \hat{\gamma}_p(\hat{\theta}_h(p; x); x) = \left(\frac{p}{1-p}\right)^2 \tilde{L}_2^x(\hat{\theta}_h(p; x)) + \tilde{L}_1^x(\hat{\theta}_h(p; x)), $$

where

$$ \tilde{L}_1^x(\hat{\theta}_h(p; x)) = \frac{\sum_{i=1}^n (Y_i - \hat{\theta}_h(p; x))^2 \delta_i (\overline{G}_n(Y_i))^{-1} \mathcal{K}_i(x)_{\{(Y_i - \hat{\theta}_h(p; x)) \le 0\}}}{\sum_{i=1}^N \mathcal{K}_i(x)}, $$

$$ \tilde{L}_2^x(\hat{\theta}_h(p; x)) = \frac{\sum_{i=1}^n (Y_i - \hat{\theta}_h(p; x))^2 \delta_i (\overline{G}_n(Y_i))^{-1} \mathcal{K}_i(x)_{\{(Y_i - \hat{\theta}_h(p; x)) > 0\}}}{\sum_{i=1}^N \mathcal{K}_i(x)}. $$

Then the confidence interval of $\theta(p; X_N)$ is obtained by

$$ \hat{\theta}_h(p; X_N) \pm U_{1-\frac{\alpha}{2}} \left( \frac{\sum_{i=1}^n \mathcal{K}_i^2(X_N) \hat{\gamma}_p(\hat{\theta}_h(p; X_N); X_N)}{\left(\sum_{i=1}^n \mathcal{K}_i(X_N)\right)^2 \overline{\sum}_p^2(\hat{\theta}_h(p; X_N), X_N)} \right)^{\frac{1}{2}}, $$

where $U_{1-\frac{\alpha}{2}}$ is the lower quantile of order $(1 - \frac{\alpha}{2})$ of the standard normal law.

**Remark 3.9** It has long been recognised that the accuracy of kernel estimators declines sharply as the dimension of the covariate space increases. In high dimensions, the local neighbourhood around any target point is typically sparsely populated unless the sample size is extraordinarily large; consequently, meaningful local averaging is impossible unless the bandwidth is taken so large that the resulting estimator becomes overly biased. This phenomenon is the celebrated *curse of dimensionality* Bellman (1961). A detailed discussion, complete with examples and numerical illustrations, is provided by Scott and Wand (1991), while more recent developments may be found in Bouzebda (2024d); Attaoui et al. (2024); Elmezouar et al. (2024); Bouzebda (2024b). A broad spectrum of dimension-reduction methods has been devised to circumvent these difficulties. The central idea is to compress the salient features of a collection of sample curves into a small set of mutually uncorrelated quantities. A natural route extends classical principal component analysis (PCA) to continuous-time stochastic processes, giving rise to *functional PCA* (FPCA) Deville (1974). The asymptotic properties of FPCA estimators have been derived in a general functional-data framework Dauxois et al. (1982), and non-parametric implementations for sparsely and irregularly observed trajectories have been developed James et al. (2000); Yao et al. (2005). As in the multivariate case, interpreting the resulting principal component scores and loadings offers valuable insight into the structure of the data. To guard against misinterpretation, Camacho et al. (2010) introduced the *Structural and Variance Information* plot. Numerical studies often suggest that explicitly penalising roughness yields more accurate estimates than FPCA. The FPCA approach tacitly assumes that the leading eigenfunctions of the predictor process–functions unrelated to the regression coefficient or the response–provide an adequate basis for representing the coefficient function. Although neither strategy dominates in all settings, methods based on reproducing-kernel Hilbert spaces (RKHS) furnish an appealing alternative Yuan and Cai (2010); Lv et al. (2018). Spline smoothing is among the most powerful and widely used techniques in non-parametric regression Eubank (1988); Wahba (1990); Green and Silverman (1994); Gu (2002). Penalised splines have gained particular popularity owing to their computational simplicity and their close connection with mixed-effects models Xiao (2019). They have also proved successful in functional data analysis–see, for instance, the *refund* package in R–yet their asymptotic theory remains technically demanding, and several fundamental questions are still unresolved even in standard regression settings. Only
<figure><img src="image_3.png" /><figcaption>Fig. 1 The functional regressor</figcaption></figure>

a handful of theoretical contributions treat penalised splines in the functional-data context Xiao (2020). Moreover, because most functional-regression procedures minimise an $L_2$ loss, they can be highly sensitive to outliers. Finally, we note that other promising approaches include techniques based on delta sequences Bouzebda et al. (2023c); Bouzebda and Nezzal (2024a), wavelet methods Didi and Bouzebda (2022), and more general methods Bouzebda and Taachouche (2025, 2024).

## 4 Monte Carlo experiments

In this section, we discuss the feasibility and performance of the constructed estimator through an empirical analysis. Precisely, our main purposes are:

(1) Prove the easy computability of the constructed estimator;

(2) Evaluate the effect of the censorship as well as the data-correlation phenomena on the performance of the estimator.

For this aim, we consider a sequence of $m$-dependent variables

$$X_i = \sum_{i}^{i+m} Z_i,$$

where $(Z_i)_i$ are independent functional random variables obtained by smoothing the curves drawn from the code routine `dgp.fiid`. These random variables form an $\alpha$-mixing process satisfying Assumption (CO5). The shapes of the functional variables of different dependency stages are shown in Figure 1.

Concerning the response variables, we use the following regression model to generate them

$$Y_i = r(X_i) + \epsilon_i, \quad i = 1, \dots, n.$$

To cover many important situations, we simulate with three different functions of $r(\cdot)$ that are:

$$r_1(X) = \int_0^1 \sin(\pi t) X(t) dt,$$
$$r_2(X) = \int_0^1 \sin(\pi t) X^2(t) dt,$$

$$r_3(X) = \int_0^1 \frac{1}{X^2(t) + 1} dt.$$

Observe that the conditional distribution of $Y$ given $X = x$ is explicitly given by the law of $\epsilon_i$ shifted by $r(X)$. So, assuming that $(\epsilon_i)_i$ is a normal distribution $N(0, 0.5)$ implies that the conditional distribution of $Y_i$ given $X_i$ is Gaussian with a conditional mean equal to $r(X_i)$. Thus, the theoretical expectile regression $\theta(p; \cdot)$ can be calculated explicitly using the routine code `genorm` in the R-package VGAM.

Recall that the main goal of this simulation experiment is to show the applicability of our estimator in practice, with special attention paid to the influence of the two principal parameters involved in this approach, such as the mixing assumption and the censoring rate. While the correlation structure is controlled by the parameter $m$, we control the influence of the censorship character by considering a censoring variable $C$ distributed as an exponential distribution $Exp(\lambda)$. Typically, the censoring rate is evaluated by the parameter $\lambda$. Exactly, we choose three values of $\lambda$ that are $\lambda = 2, 1, 0.5$. Such values allow for the generation of data with three censorship percentages 10%, 30%, and 60%. Concerning the data correlation, we also simulate three values of $m = 0, 3, 9$ (independent, medium, and strong dependency).

We point out that the simple implementation of the estimator $\hat{\theta}_{n;h}(p; \cdot)$ is closely linked to the flexibility of the choice of parameters involved in the estimator. Specifically, the practical use of the estimator $\hat{\theta}_{n;h}(p; \cdot)$ is based on the determination of the metric $d$, the bandwidth parameter $h_n$, and the kernel $K(\cdot)$. The choice of $K(\cdot)$ equals the quadratic kernel on $(0, 1)$ is adequate with the technical assumptions. Next, the metric choice is closely related to the nature of the functional variable and its smoothing property. The $L_2$ metric based on the first derivative of the functional variable is more suitable for this type of continuous functional regressor. So, all that remains is to select the bandwidth parameter $h_n$. For this last, we use the cross-validation rule based on the Asymmetric Least Squares Errors defined by

$$h_{CV} = \arg \min_{h_n \in H_n} \sum_{i=1}^{n} \rho_p(Y_i - \hat{\theta}_{n;h}(p; X_i)),$$

where $\rho_p(t) = t^2|p - \{t \le 0\}|$ and $H_n$ is the set of positive real numbers, $h_n$, such that the ball centered at $x$ with radius $h_n$ contains exactly $k$ neighbors of $x$. Typically, the number $k$ is selected from the subset $\{5, 10, 20, \dots, \frac{n}{2}\}$.

For our goal, we generate $n = 300$ observations of $(X_i, Y_i)$ and we split randomly these observations into two subsets: 200 observations in the in-sample and 100 observations in the out-sample. In Figs. 2–4, we plot the estimated conditional expectile ($p = 0.05$) of the out-sample against the theoretical one for the various treated situations. In each situation, we add a continuous line to test the estimation results in the sense that the efficiency of the estimation method is quantified by the closeness of the rounded point to this continuous line.

We want to point out that this graphical comparison corresponds only to the case of $r_3(\cdot)$. We have enhanced this comparison study by computing the averaged squared errors

$$MSE = \frac{1}{n} \sum_{i=1}^{n} (\hat{\theta}_{n;h}(p; X_i) - \theta(p; X_i))^2,$$

for various cases of $r$ and different values of $p = 0.01, 0.1, 0.5, 0.95$
<figure><img src="image_4.png" /><figcaption>Fig. 2 The estimated expectile versus the theoretical one: Independent case</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3 The estimated expectile versus the theoretical one: Medium case</figcaption></figure>

It appears clearly that the performance of both estimators is less affected by the percentage of censoring and the degree of dependence. The estimation quality is dramatically destroyed in the strong dependence and strong censoring case. However, this limitation can be effectively mitigated by increasing the sample size *n*, as illustrated in Table 3. The table presents the mean squared error (MSE) for the first model with parameters $m = 9$ and $\lambda = 0.5$, across various sample sizes ($n = 300, 500, 600$) and kernel types (Quadratic, Beta, and Gaussian). The findings indicate that kernel selection has a negligible effect on performance, whereas the MSE notably decreases as the sample size increases. Overall, the estimator demonstrates practical applicability, straightforward computational implementation, and robust performance even under conditions of strong data dependence.
<table><caption>**Table 1** MSE Results for *n* = 500</caption><thead><tr><th>Model</th><th>m</th><th>λ</th><th>p = 0.01</th><th>p = 0.1</th><th>p = 0.5</th><th>p = 0.95</th></tr></thead><tbody><tr><td rowspan="12">Model r<sub>1</sub>(·)</td><td rowspan="3">0</td><td>2</td><td>0.89</td><td>0.97</td><td>0.34</td><td>0.26</td></tr><tr><td>1</td><td>0.94</td><td>1.05</td><td>0.57</td><td>0.39</td></tr><tr><td>0.5</td><td>1.84</td><td>1.98</td><td>0.67</td><td>0.42</td></tr><tr><td rowspan="3">3</td><td>2</td><td>1.12</td><td>1.22</td><td>0.59</td><td>0.36</td></tr><tr><td>1</td><td>1.44</td><td>1.56</td><td>0.78</td><td>0.44</td></tr><tr><td>0.5</td><td>2.74</td><td>2.97</td><td>1.14</td><td>0.49</td></tr><tr><td rowspan="3">9</td><td>2</td><td>1.41</td><td>1.52</td><td>1.09</td><td>0.65</td></tr><tr><td>1</td><td>1.64</td><td>1.75</td><td>1.17</td><td>0.77</td></tr><tr><td>0.5</td><td>2.94</td><td>3.07</td><td>1.97</td><td>0.81</td></tr><tr><td rowspan="12">Model r<sub>2</sub>(·)</td><td rowspan="3">0</td><td>2</td><td>0.81</td><td>0.92</td><td>0.48</td><td>0.19</td></tr><tr><td>1</td><td>1.06</td><td>1.15</td><td>0.73</td><td>0.39</td></tr><tr><td>0.5</td><td>1.97</td><td>2.02</td><td>0.88</td><td>0.48</td></tr><tr><td rowspan="3">3</td><td>2</td><td>1.03</td><td>1.20</td><td>0.72</td><td>0.38</td></tr><tr><td>1</td><td>1.34</td><td>1.57</td><td>0.85</td><td>0.44</td></tr><tr><td>0.5</td><td>2.04</td><td>2.12</td><td>1.24</td><td>0.73</td></tr><tr><td rowspan="3">9</td><td>2</td><td>1.31</td><td>1.42</td><td>0.98</td><td>0.54</td></tr><tr><td>1</td><td>1.88</td><td>1.95</td><td>1.13</td><td>0.8</td></tr><tr><td>0.5</td><td>2.34</td><td>2.87</td><td>1.76</td><td>0.91</td></tr><tr><td rowspan="12">Model r<sub>3</sub>(·)</td><td rowspan="3">0</td><td>2</td><td>0.07</td><td>0.12</td><td>0.23</td><td>0.07</td></tr><tr><td>1</td><td>0.14</td><td>0.18</td><td>0.29</td><td>0.12</td></tr><tr><td>0.5</td><td>0.37</td><td>0.75</td><td>0.34</td><td>0.15</td></tr><tr><td rowspan="3">3</td><td>2</td><td>0.21</td><td>0.22</td><td>0.28</td><td>0.11</td></tr><tr><td>1</td><td>0.36</td><td>0.41</td><td>0.45</td><td>0.21</td></tr><tr><td>0.5</td><td>1.04</td><td>1.17</td><td>0.68</td><td>0.26</td></tr><tr><td rowspan="3">9</td><td>2</td><td>0.51</td><td>0.62</td><td>0.78</td><td>0.31</td></tr><tr><td>1</td><td>0.82</td><td>0.95</td><td>0.82</td><td>0.35</td></tr><tr><td>0.5</td><td>1.44</td><td>1.59</td><td>0.94</td><td>0.38</td></tr></tbody></table>

<table><caption>**Table 2** MSE Results for *n* = 600</caption><thead><tr><th>n</th><th>Kernel</th><th>p = 0.01</th><th>p = 0.1</th><th>p = 0.5</th><th>p = 0.95</th></tr></thead><tbody><tr><td rowspan="3">300</td><td>Quadratic</td><td>2.94</td><td>3.07</td><td>1.97</td><td>0.81</td></tr><tr><td>Beta</td><td>2.83</td><td>3.09</td><td>1.86</td><td>0.73</td></tr><tr><td>Gaussian</td><td>2.91</td><td>3.15</td><td>1.92</td><td>0.91</td></tr><tr><td rowspan="3">500</td><td>Quadratic</td><td>1.12</td><td>1.19</td><td>0.95</td><td>0.34</td></tr><tr><td>Beta</td><td>1.16</td><td>1.29</td><td>0.82</td><td>0.37</td></tr><tr><td>Gaussian</td><td>1.33</td><td>1.44</td><td>0.96</td><td>0.38</td></tr><tr><td rowspan="3">600</td><td>Quadratic</td><td>0.93</td><td>0.99</td><td>0.45</td><td>0.18</td></tr><tr><td>Beta</td><td>1.04</td><td>1.09</td><td>0.42</td><td>0.21</td></tr><tr><td>Gaussian</td><td>1.12</td><td>1.23</td><td>0.84</td><td>0.26</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Fig. 4 The estimated expectile versus the theoretical one: Strong case</figcaption></figure>

## 5 Real data example

The object of this section is to evaluate the performance of the censored functional version of the expectile kernel estimator using real financial time series data. In the last section, we have evaluated the easy implementation of the constructed estimator for different scenarios of dependency as well as censorship. In many previous contributions, we have proved the superiority of the expectile regression as a financial risk model over its competitors, such as the Value at Risk (VaR) or the Expected Shortfall (ES) model. Such superiority is mainly due to its recommended features in financial risk analysis, such as the high sensitivity to outliers, elicitability, or coherency. However, in the past few years, some authors have improved the performance of VaR and ES using some semiparametric scoring function (see, for instance, Taylor (2019) or Dimitriadis and Bayer (2019)). Even if the cited works have focused only on multivariate financial time series, their results considerably improve the efficiency of the standard models’ VaR and ES. Now, in this section, we compare these multivariate tools to our nonparametric functional approach using censored financial time series data. Recall that, from a theoretical point of view, the nonparametric functional tools are more appropriate than the parametric models to fit the high-frequency data, for instance, Ferraty and Vieu (2006). Thus, we aim in this paragraph to highlight this feature of nonparametric functional data analysis. For this purpose, we consider as a financial time series the intraday return of the stock index Nikkei during the period 11/ October/1983 to 11/October/2022. Of course, this time-varying financial data can be treated as censored time series data because, usually, trading in stock markets is stopped whenever it falls below a certain index, namely in the crisis period. Taking into account the many stops during the period of the COVID-19 pandemic, we can say that the censoring rate of this time series data is very significant. The censoring rate in our dataset is approximately 17.52%. The data under study were retrieved from the Federal Reserve Economic Data (FRED) website: https://fred.stlouisfed.org/series/NIKKEI225. As illustrated in Figure 5, the daily return series *r*(*t*) exhibits typical characteristics of financial time series, such as heavy tails, non-normality, volatility clustering, and excess kurtosis.
<figure><img src="image_7.png" /><figcaption>Fig. 5 Daily returns of the Nikkei 225 stock index</figcaption></figure>

In financial modeling, it is customary to treat time series as continuous-time processes. Here, we define the transformed process as

$$Z(t) = -100 \log \left( \frac{r(t)}{r(t-1)} \right).$$

Our objective in risk analysis is to anticipate extreme losses on a future day $d$, leveraging the historical trajectory over the past $R$ days. Conventionally, this forecasting task is formulated by setting $Y = Z(d)$ and $X(t) = Z(t_k)$ for some $t_k \in [d-R, d-1]$. However, in the framework of functional data analysis, the entire trajectory is considered, i.e., $X(t) = Z(t)$ for $t \in [d-R, d-1]$. For our comparative study, we considered different values of the historical window: $R = 3, 12$, and $90$. In the case of the Multivariate Semi-Parametric Model (MSPM), we adapted the routine from the R-package `esreg` to handle censoring. This routine is implemented using the specification functions $G_1(t) = t$ and $G_2(t) = \frac{e^t}{1+e^t}$, which are required to define the associated scoring functions (see Dimitriadis and Bayer (2019) for further details). Since such semi-parametric approaches are only applicable for a limited number of regressors, we employed Principal Component Analysis (PCA) to reduce dimensionality when $R$ is large (e.g., $R = 90$). For the Non-Parametric Functional Model (NPFM), we used the same kernel and smoothing parameter selection procedure as described in Section 4. To emphasize the functional nature of the data, we also applied smoothing using spline basis functions. The resulting smoothed trajectories are displayed in Figure 6. In this context, the spline metric is particularly appropriate; see, for example, Ferraty and Vieu (2006).

To conclude, we assess the feasibility of the three models for various probability levels: $p = 0.05, 0.10$, and $0.95$. For each fixed value of $p$, the dataset is randomly partitioned 100 times into a training set (80%) and a testing set (20%). In the subsequent tables, we report the average values of the backtesting metric proposed by Ferraty and Quintela-del Río (2016),
<figure><img src="image_8.png" /><figcaption>Fig. 6 The smoothed curves for the case when R = 90</figcaption></figure>

<table><caption>**Table 3** BTM Results for different values of p</caption><thead><tr><th>Mode</th><th>R</th><th>p = 0.1</th><th>p = 0.05</th><th>p = 0.95</th></tr></thead><tbody><tr><td rowspan="3">VaR</td><td>3</td><td>0.94</td><td>1.03</td><td>0.81</td></tr><tr><td>12</td><td>1.62</td><td>1.34</td><td>0.73</td></tr><tr><td>90</td><td>2.51</td><td>2.67</td><td>0.95</td></tr><tr><td rowspan="3">ES</td><td>3</td><td>0.82</td><td>1.07</td><td>0.75</td></tr><tr><td>12</td><td>2.42</td><td>3.09</td><td>0.83</td></tr><tr><td>90</td><td>2.23</td><td>3.15</td><td>0.92</td></tr><tr><td rowspan="3">Expectile S</td><td>3</td><td>1.07</td><td>1.12</td><td>0.68</td></tr><tr><td>12</td><td>1.58</td><td>1.23</td><td>0.59</td></tr><tr><td>90</td><td>1.29</td><td>1.37</td><td>0.66</td></tr></tbody></table>

defined as:

$$BTM = \frac{1}{N} \sum_{i=1}^{N} (Y_i - \hat{\zeta}_{n;h}(p; X_i)) \mathbb{1}_{\{Y_i > \zeta(p; X_i)\}},$$

where

$$N = \sum_{j \in \text{testing sample}} \mathbb{1}_{\{Y_j > \zeta(p; X_j)\}}.$$

Here, $\zeta(p; X_i)$ denotes either the functional estimator of the expectile or the MSPM estimators of Value-at-Risk (VaR) and Expected Shortfall (ES), depending on the context.

The exact values of the different replications are displayed in the following box-plot figures. It concerns the three models for $p = 0.1$. The results are in Figs. 7–9.

Without surprise, we can see that the different test measures are strongly linked to the dimensionality of the data. In the sense that the multivariate approach has good performance for small $R = 3$. While the nonparametric functional procedure is a good solution for the high-frequency data associated with a large $R = 90$. All in all, it seems that the expectile regression behaves more accurately with the outliers, and then to the extreme loss. This
<figure><img src="image_9.png" /><figcaption>Fig. 7 The BTM in the case when $R = 3$</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8 The BTM in the case when $R = 12$</figcaption></figure>

statement is confirmed by comparing the interquartile range (IQR) of different BTM errors with different values of $R$. It appears that the IQR of the expectile is more important than the other models.

## 6 Concluding remarks

In this paper, we have investigated the problem of nonparametric estimation of the expectile regression model for strong mixing censored functional data. Consideration is given to the kernel-type estimator for the expectile regression, and strong almost-complete consistency is obtained with its rate and asymptotic normality. The effectiveness of the proposed methodology is demonstrated by the results of a finite sample and a sample of actual data. It will be natural to consider in a future investigation of the functional $k$NN local linear approach expectile regression estimators to obtain an alternative estimator that benefits from
<figure><img src="image_11.png" /><figcaption>Fig. 9 The BTM in the case when R = 90</figcaption></figure>

the advantages of both methods, the local linear method and the kNN approach. Observe that mixing is a type of asymptotic independence assumption that is commonly used in the pursuit of simplicity, but can be unrealistic in situations where the data are highly dependent. Extending non-parametric functional ideas to a general dependence structure is a discipline that is still in the beginning stages. Notably, the ergodic framework eschews the commonly employed strong mixing condition and its variants for measuring dependence, as well as the extremely involved probabilistic calculations that this condition necessitates. It would be intriguing to extend our work to the case of functional ergodic data, which requires nontrivial mathematics; however, this is well outside the scope of this paper. As a perspective on this work, one can relax the stationarity assumption we made in this paper and investigate similar limit theorems for local stationary functional ergodic processes. In this case, we allow the conditional expectile estimator to change smoothly over time. It would be useful to establish similar results in this paper for data-driven bandwidth. The proof of such a statement, however, should require a different methodology than that used in the present paper, and we leave this problem open for future research.

## 7 Proofs of Theorems

This section is devoted to the proof of our main result. The previously presented notation continues to be used in the following.

### Proof of Theorem 3.8

Let us introduce

$$U_n = \left( \frac{n\phi_x(h)}{\sigma_p^2(x)} \right)^{\frac{1}{2}} (\hat{\theta}_{n,h_n}(p; x) - \theta(p; x)).$$
For $u \in \mathbb{R}$, we set

$$v_p(u, x) = \theta(p; x) + u(n\phi_x(h))^{-\frac{1}{2}}\sigma_p(x).$$

We get the following decomposition:

$$ 
\begin{aligned}
\mathbb{P}(U_n \le u) &= \mathbb{P}(\hat{\theta}_{n,h}(p; x) \le v_p(u, x)) \\
&= \mathbb{P}(\{\hat{\theta}_{n,h}(p; x) \le v_p(u, x)\} \cap \{\hat{\mathcal{E}}_{n;h;2}(v_p(u, x)) = 0\}) \\
&\quad + \mathbb{P}(\{\hat{\theta}_{n,h}(p; x) \le v_p(u, x)\} \cap \{\hat{\mathcal{E}}_{n;h;2}(v_p(u, x)) \ne 0\}) \\
&= U_1(x) + U_2(x).
\end{aligned} 
\quad (7.1) $$

Remark that, for the first term $U_1(x)$ of equation 7.1, we have

$$ 
\begin{aligned}
U_1(x) &= \mathbb{P}(\{\hat{\theta}_{n,h}(p; x) \le v_p(u, x)\} \cap \{\hat{\mathcal{E}}_{n;h;2}(v_p(u, x)) = 0\}) \\
&\le \mathbb{P}(\{\hat{\mathcal{E}}_{n;h;2}(v_p(u, x)) = 0\}).
\end{aligned} 
$$

For the second term $U_2(x)$, and by using the fact that the function $\hat{\mathcal{E}}_{n;h}(\cdot; x)$ is an increasing function and $\hat{\theta}_{n,h}(p; x)$ is the unique solution of the equation $\hat{\mathcal{E}}_{n,h}(\theta(p; x); x) = \frac{p}{1-p}$, we obtain

$$ 
\begin{aligned}
\{\hat{\theta}_{n,h}(p; x) \le v_p(u, x)\} &\iff \left\{\frac{p}{1-p} \le \hat{\mathcal{E}}_{n;h}(v_p(u; x); x)\right\} \\
&\iff \left\{\frac{p}{1-p} \le \frac{\hat{\mathcal{E}}_{n;h;1}(v_p(u, x); x)}{\hat{\mathcal{E}}_{n;h;2}(v_p(u, x); x)}\right\}.
\end{aligned} 
$$

The term $U_2(x)$ can be written as follows:

$$ 
\begin{aligned}
U_2(x) &= \mathbb{P}\left(\frac{p}{1-p} \le \frac{\hat{\mathcal{E}}_{n;h;1}(v_p(u, x); x)}{\hat{\mathcal{E}}_{n;h;2}(v_p(u, x); x)}\right) \\
&= \mathbb{P}\left(\left(\frac{p}{1-p}\right) \hat{\mathcal{E}}_{n;h;2}(v_p(u, x); x) - \hat{\mathcal{E}}_{n;h;1}(v_p(u, x); x) \le 0\right),
\end{aligned} 
$$

we have

$$ U_2(x) = \mathbb{P}(\hat{\Xi}_p(u, x) - \mathbb{E}[\hat{\Xi}_p(u, x)] \le \mathbb{E}[-\hat{\Xi}_p(u, x)]), \quad (7.2) $$

where

$$ \hat{\Xi}_p(u, x) = \left(\frac{p}{1-p}\right) \hat{\mathcal{E}}_{n;h;2}(v_p(u, x)) - \hat{\mathcal{E}}_{n;h;1}(v_p(u, x)). $$

Lemmas 7.1-7.3 below are oriented towards proving Theorem 3.8.

**Lemma 7.1** Under the conditions (CO3) and (NA1)-(NA3), we have, as $n \to \infty$,

$$ U_1(x) \to 0, \quad (7.3) $$

where $U_1(x)$ is defined in Eq. (7.1).

**Lemma 7.2** Assume that the assumptions of Theorem 3.8 are satisfied. We have, as $n \to \infty$,

$$ \left(\frac{n\phi_x(h)}{\sigma_p^2(x)}\right)^{\frac{1}{2}} \left(\sum_p (\theta(p; x); x)\right)^{-1} \mathbb{E}[-\hat{\Xi}_p(u, x)] = u + o(1). \quad (7.4) $$
**Lemma 7.3** Suppose that the assumptions of Theorem 3.8 hold. We have, as $n \to \infty$,

$$ U_n = \left( \frac{n\phi_x(h)}{\sigma_p^2(x)} \right)^{\frac{1}{2}} \left( \Sigma_p (\theta(p; x); x) \right)^{-1} (\hat{\Xi}_p(u, x) - \mathbb{E}[\hat{\Xi}_p(u, x)]) \xrightarrow{D} N(0, 1), \quad (7.5) $$

where

$$ \hat{\Xi}_p(u, x) = \left( \frac{p}{1-p} \right) \hat{\mathcal{E}}_{n;h;2}(\nu_p(u, x)) - \hat{\mathcal{E}}_{n;h;1}(\nu_p(u, x)), $$

and

$$ (n\phi_x(h)) \mathrm{Var} \left( \sum_{i=1}^{n} \mu_i(x) \right) \rightarrow \frac{\alpha_2(x)\gamma_p(\theta(p; x); x)}{\alpha_1^2(x)}, \quad (7.6) $$

where

$$ \mu_i(x) = \frac{1}{n\mathbb{E}[K_1(x)]} (\mathbb{B}_i - \mathbb{E}[\mathbb{B}_i]), $$

and

$$ \mathbb{B}_i = K_i(x)\delta_i ((\bar{\mathbf{G}}_n(Y_i))^{-1}) \left( \frac{p}{1-p} \psi_2(Y_i - \nu_p(u, x)) + \psi_1(Y_i - \nu_p(u, x)) \right). $$

**Proof of Lemma 3.3**

By using the strong law of large numbers, for $s, l = 1, 2$, and the fact that

$$ \mathbb{1}_{\{T_i \le C_i\}} \psi_l(Y_i - z) = \mathbb{1}_{\{T_i \le C_i\}} \psi_l(T_i - z), $$

we get

$$ \begin{aligned} |\hat{\mathcal{E}}_{n;h;s}(z; x) - \tilde{\mathcal{E}}_{n;h;s}(z; x)| &= \frac{1}{n\mathbb{E}[K_1(x)]} \left| \sum_{i=1}^{n} \mathcal{K}_i(x) \delta_i \psi_l(Y_i - z) \left( (\bar{\mathbf{G}}_n(Y_i))^{-1} - (\bar{\mathbf{G}}(Y_i))^{-1} \right) \right| \\ &\le \frac{1}{n\mathbb{E}[K_1(x)]} \sum_{i=1}^{n} \mathcal{K}_i(x) \mathbb{1}_{\{T_i \le C_i\}} \psi_l(T_i - z) \left| (\bar{\mathbf{G}}_n(T_i))^{-1} - (\bar{\mathbf{G}}(T_i))^{-1} \right| \\ &\le \sup_{t \le \tau_F} \frac{|\bar{\mathbf{G}}_n(t) - \bar{\mathbf{G}}(t)|}{\bar{\mathbf{G}}_n(\tau_F)\bar{\mathbf{G}}(\tau_F)} \frac{1}{n\mathbb{E}[K_1(x)]} \sum_{i=1}^{n} \mathcal{K}_i(x) \mathbb{1}_{\{T_i \le C_i\}} \psi_l(T_i - z) \\ &\le \frac{|\tau_F - z|}{\bar{\mathbf{G}}_n(\tau_F)\bar{\mathbf{G}}(\tau_F)} \sup_{t \le \tau_F} \frac{|\bar{\mathbf{G}}_n(t) - \bar{\mathbf{G}}(t)|}{n\mathbb{E}[K_1(x)]} \sum_{i=1}^{n} |\mathcal{K}_i(x)| \\ &\le \frac{|\tau_F - z|}{\bar{\mathbf{G}}_n(\tau_F)\bar{\mathbf{G}}(\tau_F)} \sup_{t \le \tau_F} \frac{|\bar{\mathbf{G}}_n(t) - \bar{\mathbf{G}}(t)|}{\mathbb{E}\left[\frac{1}{\mathbb{E}[K_1(x)]}\mathcal{K}_i(x)\right]}, \end{aligned} $$

under conditions (CO1), (CO4) and the law of iterated logarithm from formula (4.28) in Deheuvels and Einmahl (2000), see also Földes and Rejtő (1981), we obtain for $s = 1, 2$

$$ |\hat{\mathcal{E}}_{n;h;s}(z; x) - \tilde{\mathcal{E}}_{n;h;s}(z; x)| \le \frac{|\tau_F - z|}{\bar{\mathbf{G}}_n(\tau_F)\bar{\mathbf{G}}(\tau_F)} \sqrt{\frac{\log \log(n)}{n}} \mathbb{E} \left[ \left| \frac{1}{\mathbb{E}[K_1(x)]} \mathcal{K}_i(x) \right| \right]. $$

We readily infer, for $s = 1, 2$

$$ |\hat{\mathcal{E}}_{n;h;s}(z; x) - \tilde{\mathcal{E}}_{n;h;s}(z; x)| = O_{a.s.} \left( \sqrt{\frac{\log \log(n)}{n}} \right). $$
This suffices to prove Lemma 3.3.

□

**Proof of Lemma 3.4**

By using double conditioning with respect to $T_i$, we get

$$
\begin{align*}
\mathbb{E}[\tilde{\mathcal{E}}_{n;h;1}(z; x)] &= \frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} \sum_{i=1}^{n} \mathbb{E}[\mathbb{E}(\delta_i (\overline{\mathbf{G}}(Y_i))^{-1} \psi_1(Y_i - z) \mathcal{K}_i(x) | X_i)] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}[\mathcal{K}_1(x) (\mathbb{E}(\delta_1 (\overline{\mathbf{G}}(Y_1))^{-1} \psi_1(Y_1 - z) | X_1))] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}[\mathcal{K}_1(x) \mathbb{E}(\mathbb{E}[\delta_1 (\overline{\mathbf{G}}(Y_1))^{-1} \psi_1(Y_1 - z) | T_1]) | X_1] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}[\mathcal{K}_1(x) \mathbb{E}[\overline{\mathbf{G}}(T_1))^{-1} \psi_1(T_1 - z) \mathbb{E}(1_{\{T_1 \le C_1\}} | T_1) | X_1]] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}[\mathcal{K}_1(x) \mathbb{E}[\overline{\mathbf{G}}(T_1))^{-1} \psi_1(T_1 - z) 1_{\{T_1 \le C_1\}} | X_1]].
\end{align*}
$$

Then, we readily infer

$$|\mathbb{E}[\tilde{\mathcal{E}}_{n;h;1}(z; x)] - \mathcal{E}_1(z; x)| = \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}[\mathcal{K}_1(x) 1_{B(x,h)}(X_1) | \mathcal{E}_1(z; X_1) - \mathcal{E}_1(z; x)],$$

thus, under condition (CO2) for $i = 1$, we get

$$|\mathbb{E}[\tilde{\mathcal{E}}_{n;h;1}(z; x)] - \mathcal{E}_1(z; x)| \le C h^{k_1} = O(h^{k_1}).$$

To evaluate (3.6), we use the same arguments as those involved in the proof of (3.5). We get, under conditions (CO2) for $i = 2$,

$$
\begin{align*}
|\mathcal{E}_2(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;2}(z; x)]| &= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}[\mathcal{K}_1(x) 1_{B(x,h)}(X_1) | \mathcal{E}_2(z; x) - \mathcal{E}_2(z; X_1)] \\
&\le C h^{k_2} = O(h^{k_2}).
\end{align*}
$$

We obtain the result of Lemma 3.4.

□

**Proof of Lemma 3.5**

For this lemma, $Y$ is not necessarily bounded, we use the truncation procedure and introduce the following function:

$$\tilde{\mathcal{E}}_{n;h;s}^*(z; x) = -\frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} \sum_{i=1}^{n} \delta_i (\overline{\mathbf{G}}(Y_i))^{-1} \mathcal{K}_i(x) \psi^*(Y_i - z),$$

where

$$\psi^* = \psi 1_{\{|\psi| \le \gamma_n|},}$$

with $\gamma_n$ is the sequence which tends to infinity as $n$ tends to infinity. The proof of the results 3.7 and 3.8 are based on the following inequality: for $s = 1, 2$

$$|\tilde{\mathcal{E}}_{n;h;s}(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}(z; x)]| \le |\tilde{\mathcal{E}}_{n;h;s}^*(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}^*(z; x)]| + |\tilde{\mathcal{E}}_{n;h;s}(z; x) - \tilde{\mathcal{E}}_{n;h;s}^*(z; x)|$$
$$+ |\mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}^*(z; x)] - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}(z; x)]| \\ = R_{33} + R_{32} + R_{31}.$$

So, for the term $R_{31}$, notice that we have

$$\delta_i (\bar{\mathbf{G}}(Y_i))^{-1} = \mathbb{1}_{\{T_i \le C_i\}} (\bar{\mathbf{G}}(T_i))^{-1} = \mathbb{E}[\mathbb{1}_{\{T_i \le C_i\}} (\bar{\mathbf{G}}(T_i))^{-1} | T_i] \\ = (\bar{\mathbf{G}}(T_i))^{-1} \mathbb{P}(\{T_i \le C_i\} | T_i) = (\bar{\mathbf{G}}(T_i))^{-1} \bar{\mathbf{G}}(T_i) = 1. \quad (7.7)$$

Then, for $s = 1, 2$ we have

$$R_{31} = |\mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}^*(z; x)] - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}(z; x)]| \\ \le \frac{C}{\phi_x(h)} \mathbb{E}[|\psi| \delta_i (\bar{\mathbf{G}}(Y_i))^{-1} \mathcal{K}_i(x) \mathbb{1}_{\{|\psi| > \gamma_n\}}] \\ \le \frac{C}{\phi_x(h)} \mathbb{E}[|\psi| \mathcal{K}_1(x) \mathbb{1}_{\{|\psi| > \gamma_n\}}]. \quad (7.8)$$

By applying Hölder's and Markov's inequality with $\alpha = \frac{1}{2}$ and $\beta$ such that: $\frac{1}{\alpha} + \frac{1}{\beta} = 1$, by (CO1) and (CO4), we get

$$R_{31} \le \frac{C}{\phi_x(h)} \mathbb{E}^{\frac{1}{\alpha}} [|\psi|^\alpha \mathbb{1}_{\{|\psi| > \gamma_n\}}] \mathbb{E}^{\frac{1}{\beta}} [\mathcal{K}_1^\beta(x)] \\ \le \frac{C}{\phi_x(h)} \left( \mathbb{E}^{\frac{1}{2\alpha}} [|\psi|^{2\alpha}] (\mathbb{P}(|\psi| > \gamma_n))^{\frac{1}{2\alpha}} \right) \mathbb{E}^{\frac{1}{\beta}} [\mathcal{K}_1^\beta(x)] \\ \le \frac{C}{\phi_x(h)} \left( \mathbb{E}[|\psi|^l] \frac{\mathbb{E}[|\psi|^l]}{\gamma_n^l} \right)^{\frac{1}{l}} \mathbb{E}^{\frac{1}{\beta}} [\mathcal{K}_1^\beta(x)] \\ \le \frac{C}{\phi_x(h)} \gamma_n^{-1} \mathbb{E}^{\frac{1}{\alpha}} [|\psi|^l] \phi_x^{\frac{1}{\beta}}(h) \\ \le C \gamma_n^{-1} \phi_x^{\frac{1}{\beta}-1}(h). \quad (7.9)$$

For the second term $R_{32}$, under the condition (CO3) and for $s = 1, 2$, it follows immediately from Markov's inequality that for all $\epsilon > 0$,

$$\mathbb{P}[|\tilde{\mathcal{E}}_{n;h;s}(z; x) - \tilde{\mathcal{E}}_{n;h;s}^*(z; x)| > \epsilon] = \mathbb{P}\left[\frac{1}{n\mathbb{E}[\mathcal{K}_1(x)]} \left|\sum_{i=1}^n \psi \mathbb{1}_{\{|\psi| > \gamma_n\}} \mathcal{K}_i(x)\right| > \epsilon\right] \\ \le \sum_{i=1}^n \mathbb{P}[|\psi| > \gamma_n] \\ \le n \mathbb{P}[|\psi| > \gamma_n] \\ \le n \frac{\mathbb{E}[|\psi|^l]}{\gamma_n^l} \le C n \gamma_n^{-l}.$$

In the proof of the term $R_{33}$, denote $\mathbb{L}_i(x) = A_i(x) - \mathbb{E}[A_i(x)]$ where $A_i(x) = \delta_i (\bar{\mathbf{G}}(Y_i))^{-1} \mathcal{K}_i(x) \psi^*(Y_i - z)$. Thus, for all $\epsilon > 0$, we have

$$\mathbb{P}[|\tilde{\mathcal{E}}_{n;h;\pm}^*(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}^*(z; x)]| > \epsilon] = \mathbb{P}\left[\left|\frac{1}{n\mathbb{E}[\mathcal{K}_1(x)]} \sum_{i=1}^n \mathbb{L}_i(x)\right| > \epsilon\right]$$
$$= \mathbb{P}\left[\left|\sum_{i=1}^{n} \mathbb{L}_i(x)\right| > \epsilon n \mathbb{E}[\mathcal{K}_1(x)]\right].$$

The evaluation of this quantity is based on the asymptotic behaviour of the asymptotic variance defined by

$$\begin{aligned} S_n^2 &= \sum_{i=1}^{n} \sum_{j=1}^{n} \operatorname{Cov}(\mathbb{L}_i(x), \mathbb{L}_j(x)) \\ &= S_n^{*2} + n \operatorname{Var}[\mathbb{L}_1(x)], \end{aligned} \quad (7.10)$$

where

$$S_n^{*2} = \sum_{i=1}^{n} \sum_{i \neq j} \operatorname{Cov}(\mathbb{L}_i(x), \mathbb{L}_j(x)).$$

We use Masry's techniques in Masry (2005), considering a sequence of natural numbers $\nu_n$, we split the sum into two sets defined by

$$\begin{aligned} S_1 &= \{(i, j) \text{ such that } 1 \le |i - j| \le \nu_n\} \text{ and} \\ S_2 &= \{(i, j) \text{ such that } \nu_n + 1 \le |i - j| \le n - 1\}. \end{aligned} \quad (7.11)$$

The first term of 7.10 decomposes into two sums $M_{1,n}$ and $M_{2,n}$; the sum of covariance over $S_1$ and $S_2$ respectively. On the set $S_1$, under the conditions (CO3) and (CO6), we have

$$\begin{aligned} M_{1,n} &= \sum_{S_1} |\operatorname{Cov}(\mathbb{L}_i(x), \mathbb{L}_j(x))| \\ &\le C \sum_{S_1} \left| \mathbb{E} \left[ \mathcal{K}_i(x) \mathcal{K}_j(x) \frac{\delta_i \delta_j}{\bar{\mathbf{G}}(Y_i) \bar{\mathbf{G}}(Y_j)} \right] \right| + \left| \mathbb{E} \left[ \mathcal{K}_i(x) \frac{\delta_i}{\bar{\mathbf{G}}(Y_i)} \right] \mathbb{E} \left[ \mathcal{K}_j(x) \frac{\delta_j}{\bar{\mathbf{G}}(Y_j)} \right] \right|. \end{aligned} \quad (7.12)$$

For the right-hand side of inequality 7.12, we have

$$\begin{aligned} \mathbb{E} \left[ \mathcal{K}_i(x) \frac{\delta_i}{\bar{\mathbf{G}}(Y_i)} \right] &= \mathbb{E} \left[ \mathcal{K}_i(x) \mathbb{E} \left[ \frac{\delta_i}{\bar{\mathbf{G}}(Y_i)} \mid T_i \right] \right] \\ &= \mathbb{E} \left[ \mathcal{K}_i(x) \mathbb{E} \left[ \frac{1_{\{T_i \le C_i\}}}{\bar{\mathbf{G}}(T_i)} \mid T_i \right] \right] = \mathbb{E} [\mathcal{K}_i(x)], \end{aligned} \quad (7.13)$$

and by using double conditioning with respect to $T_i$, we find that we have

$$\begin{aligned} \mathbb{E}[\delta_i \delta_j | T_i, T_j] &= \mathbb{E}[1_{\{T_i \le C_i\}} 1_{\{T_j \le C_j\}} | T_i, T_j] \\ &= \mathbb{P}(\{T_i \le C_i\} \cap \{T_j \le C_j\} | T_i, T_j) \\ &= \mathbb{P}(\{T_i \le C_i\} \cap \{T_j \le C_j\}) \\ &= \mathbb{P}(T_i \le C_i) \mathbb{P}(T_j \le C_j | T_i \le C_i) \\ &= \bar{\mathbf{G}}(T_i) \bar{\mathbf{G}}(T_j). \end{aligned} \quad (7.14)$$

From 7.14, we have

$$\mathbb{E} \left[ \mathcal{K}_i(x) \mathcal{K}_j(x) \frac{\delta_i \delta_j}{\bar{\mathbf{G}}(Y_i) \bar{\mathbf{G}}(Y_j)} \right] = \mathbb{E} \left[ \mathcal{K}_i(x) \mathcal{K}_j(x) \mathbb{E} \left[ \frac{\delta_i \delta_j}{\bar{\mathbf{G}}(Y_i) \bar{\mathbf{G}}(Y_j)} \mid T_i, T_j \right] \right]$$
$$ 
\begin{aligned}
&= \mathbb{E} \left[ \mathcal{K}_i(x) \mathcal{K}_j(x) \mathbb{E} \left[ \frac{1_{\{T_i \le C_i\}} 1_{\{T_j \le C_j\}}}{\overline{\mathbf{G}}(T_i) \overline{\mathbf{G}}(T_j)} \middle| T_i, T_j \right] \right], \\
&= \mathbb{E} \left[ \mathcal{K}_i(x) \mathcal{K}_j(x) \right],
\end{aligned} 
\quad (7.15) 
$$

which implies that under the conditions (CO1), the second part of (CO5) and (CO6), we have

$$ M_{1,n} \le C n \nu_n \chi_x(h). $$

On the other hand, by using the fact that

$$ \sum_{k \ge \nu_n+1} k^{-a} \le \int_{\nu_n}^{+\infty} t^{-a} dt $$

and the Davydov-Rio inequality (Lemma 1.2.5 of Rio (2017)) makes it possible to increase this covariance

$$ |\mathrm{Cov}(\mathbb{L}_i(x), \mathbb{L}_j(x))| \le C \gamma_n^2 \alpha(|i-j|). $$

Hence, we have

$$ M_{2,n} = \sum_{S_2} |\mathrm{Cov}(\mathbb{L}_i(x), \mathbb{L}_j(x))| \le C \frac{n \gamma_n^2 \nu_n^{-a+1}}{a-1}, $$

which translates to the following expression

$$ M_{2,n} = O(n \gamma_n^2 \nu_n^{-a+1}). $$

By choosing

$$ \nu_n = \left( \frac{\gamma_n^2}{\chi_x(h)} \right)^{1/a}, $$

we prove that

$$ \sum_{i=1}^{n} \sum_{i \neq j} \mathrm{Cov}(\mathbb{L}_i(x), \mathbb{L}_j(x)) = O \left( n \gamma_n^{2/a} \chi_x^{(a-1)/a}(h) \right). $$

On the other hand, the second term of (7.10) under the condition (CO3), we have

$$ \mathrm{Var}(\mathbb{L}_1(x)) \le \mathbb{E} \left[ \delta_i \left( \overline{\mathbf{G}}(Y_i) \right)^{-1} \mathcal{K}_i(x) \psi^*(Y_i - z) \right]^2 = O(\phi_x(h)). $$

Thus, under the condition (CO6), we get

$$ S_n^2 = O \left( n \chi_x^{(1/2)}(h) \right). \quad (7.16) $$

Now, for $s = 1, 2$, we apply Fuck-Nagaev's inequality for $\mathbb{L}_i(x)$. We have, for all $\ell > 0$ and for all $\varepsilon > 0$,

$$ 
\begin{aligned}
& \mathbb{P} \left\{ |\tilde{\mathcal{E}}_{n;h;s}^*(z; x) - \mathbb{E}[\tilde{\mathcal{E}}_{n;h;s}^*(z; x)]| > \varepsilon \right\} \le \mathbb{P} \left\{ \left| \sum_{i=1}^n \mathbb{L}_i(x) \right| > \varepsilon n \mathbb{E}[\mathcal{K}_1(x)] \right\} \\
& \le C(N_1(x) + N_2(x)),
\end{aligned}
$$
where

$$N_1(x) = \left(1 + \frac{\varepsilon^2 n^2 (\mathbb{E}[K_1(x)])^2}{S_n^2 \ell}\right)^{-\ell/2} \quad \text{and} \quad N_2(x) = n\ell^{-1} \left(\frac{\ell}{\varepsilon n \mathbb{E}[K_1(x)]}\right)^{a+1}.$$

By choosing

$$\varepsilon = \varepsilon_0 \frac{\sqrt{n \log n} \chi_x^{1/2}(h)}{n \mathbb{E}[K_1(x)]} \quad \text{and} \quad \ell = C(\log n)^2,$$

and making use of (CO6), we readily obtain

$$\begin{aligned} N_2(x) &\le C \epsilon_0^{-(a+1)} n^{1-(a+1)/2} \chi_x(h)^{-(a+1)/4} (\log n)^{(3a-1)/2} \\ &\le C n^{-1-\nu_1}, \end{aligned} \quad (7.17)$$

for some $\nu_1 > 0$. Using the same reasoning to obtain (7.17), we show, for $\ell = O(\log n)^2$, that

$$N_1(x) \le C \left( 1 + \frac{\epsilon_0^2 \log n}{\ell} \right)^{-\ell/2} \le C n^{-1-\nu_2}, \quad (7.18)$$

for some $\nu_2 > 0$. Then, the combination of the relations (7.17) and (7.18) permits to conclude (3.7, 3.8). □

### Proof of Lemma 7.1

Based on the Lemmas 3.3, 3.4 and 3.5, for all $0 < \varepsilon \le \varepsilon_2(\nu_p(u, x); x)$, we have

$$\begin{aligned} \mathbb{P}[\hat{\varepsilon}_{n;h;2}(\nu_p(u, x)) = 0] &\le \mathbb{P}[\hat{\varepsilon}_{n;h;2}(\nu_p(u, x)) \le \varepsilon_2(\nu_p(u, x); x) - \varepsilon] \\ &\le \mathbb{P}[|\hat{\varepsilon}_{n;h;2}(\nu_p(u, x)) - \varepsilon_2(\nu_p(u, x); x)| \ge \varepsilon]. \end{aligned} \quad (7.19)$$

The result is then a consequence of (3.4), (3.6) and (3.8). □

### Proof of Lemma 7.2

By the fact that $\theta(p; x)$ is the unique solution of the Eq. (2.2), we infer that

$$\varepsilon_1(\theta(p; x)) - \frac{p}{1-p} \varepsilon_2(\theta(p; x); x) = 0.$$

We have the following decomposition

$$\begin{aligned} \mathbb{E}[-\hat{\Xi}_p(u, x)] &= \mathbb{E}[\hat{\varepsilon}_{n;h;1}(\nu_p(u, x))] - \left(\frac{p}{1-p}\right) \mathbb{E}[\hat{\varepsilon}_{n;h;2}(\nu_p(u, x))] \\ &= \left(\mathbb{E}[\hat{\varepsilon}_{n;h;1}(\nu_p(u, x); x)] - \varepsilon_1(\nu_p(u, x); x)\right) \\ &\quad + \frac{p}{1-p} \left(\varepsilon_2(\nu_p(u, x); x) \mathbb{E}[\hat{\varepsilon}_{n;h;2}(\nu_p(u, x); x)]\right) \\ &\quad + \left(\varepsilon_1(\nu_p(u, x); x) - \varepsilon_1(\theta(p; x); x)\right) \\ &\quad - \frac{p}{1-p} \left(\varepsilon_2(\nu_p(z, x); x) - \varepsilon_2(\theta(p; x); x)\right) \\ &= A_1(x) + A_2(x) + A_3(x) + A_4(x). \end{aligned} \quad (7.20)$$
Making use of the conditions (CO2) (for $k > \min(k_1, k_2)$), (CO4) and (NA1), we obtain the following bound for $j_i = 0, 1$ and for $s = 1, 2$,

$$|A_i(x)| = \left(\frac{p^{j_i}}{1-p^{j_i}}\right) |\mathbb{E}[\mathcal{E}_{n;h;s}(v_p(u, x))] - \mathcal{E}_i(v_p(u, x); x)| \le C_i h^k,$$

which implies, for $i = 1, 2$,

$$|A_i(x)| = O(h^k).$$

To evaluate the terms $A_j(x)$ for $j = 3, 4$ in the right side of (7.20), we start with Taylor's expansion of the function $\mathcal{E}_i(\cdot; x)$, for $i=1,2$ and $l_i = 0, 1$, to obtain

$$\begin{aligned} A_j(x) &= \mathcal{E}_i(v_p(u, x); x) - \mathcal{E}_i(\theta(p; x); x) \\ &= u\sigma_p(x) (n\phi_x(h))^{-\frac{1}{2}} \left(-\frac{p^{l_i}}{1-p^{l_i}}\right) \mathcal{E}_i'(\theta(p; x); x) + o((n\phi_x(h))^{-\frac{1}{2}}). \end{aligned}$$

Then, by setting

$$\Sigma_p(\theta(p; x); x) = \mathcal{E}'_1(\theta(p; x); x) - \left(\frac{p}{1-p}\right) \mathcal{E}'_2(\theta(p; x); x),$$

one can see that

$$\left|\sum_{i=1}^{4} A_i(x)\right| = O(h^k) + u\sigma_p(x) (n\phi_x(h))^{-\frac{1}{2}} \left(\Sigma_p(\theta(p; x); x)\right) + o\left((n\phi_x(h))^{-\frac{1}{2}}\right).$$

Under the third part of the assumption (NA4), for $k > \min(k_1, k_2)$, we get the following result

$$\begin{aligned} \left(\frac{n\phi_x(h)}{\sigma_p^2(x)}\right)^{\frac{1}{2}} \left(\Sigma_p(\theta(p; x); x)\right)^{-1} \left|\mathbb{E}\left(-\hat{\Xi}_p(u, x)\right)\right| &= O\left(nh_n^{2k}\phi_x(h)\right)^{\frac{1}{2}} + u + o(1) \\ &= u + o(1). \end{aligned}$$

Therefore, the proof of Lemma 7.2 is completed. □

### Proof of Lemma 7.3

Let

$$\mathrm{Var}\left(\sum_{i=1}^{n} \mu_i(x)\right) = \sum_{i=1}^{n} \mathrm{Var}(\mu_i(x)) + 2 \sum_{i \neq j} \mathrm{Cov}[\mu_i(x), \mu_j(x)], \quad (7.21)$$

the asymptotic variance of $\hat{\Xi}_p(u, x)$ defined in (7.6) is consequence of the two following statements:

$$\sum_{i=1}^{n} \mathrm{Var}(\mu_i(x)) = \frac{\alpha_2(x)\gamma_p(\theta(p; x); x)}{n\phi_x(h)\alpha_1^2(x)} + o\left(\frac{1}{n\phi_x(h)}\right), \quad (7.22)$$

where

$$\gamma_p(u; x) = \left(\frac{p}{1-p}\right)^2 L_2^x(u) + L_1^x(u),$$
with

$$L_2^x(u) = \mathbb{E}(\psi_2^2(Y_1 - u) | X = x), \quad L_1^x(u) = \mathbb{E}(\psi_1^2(Y_1 - u) | X = x),$$

and

$$\sum_{i \neq j} \mathrm{Cov}[\mu_i(x), \mu_j(x)] = o\left(\frac{1}{n\phi_x(h)}\right). \qquad (7.23)$$

For the first statement, we set the following quantity:

$$\mu_i(x) = \frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} (\mathbb{B}_i(x) - \mathbb{E}[\mathbb{B}_i(x)]),$$

with

$$\mathbb{B}_i(x) = \mathcal{K}_i(x) \delta_i(\overline{\mathbf{G}}_n(Y_i))^{-1} \left( \frac{p}{1-p} \psi_2(Y_i - v_p(u, x)) + \psi_1(Y_i - v_p(u, x)) \right).$$

We have

$$\begin{aligned}
\sum_{i=1}^{n} \mathrm{Var}(\mu_i(x)) &= \sum_{i=1}^{n} \mathrm{Var} \left[ \frac{1}{n \mathbb{E}[\mathcal{K}_1(x)]} (\mathbb{B}_i - \mathbb{E}[\mathbb{B}_i]) \right] \\
&= \frac{1}{n \mathbb{E}^2[\mathcal{K}_1(x)]} \mathrm{Var}(\mathbb{B}_1) \\
&= \frac{1}{n} \left( \frac{\mathbb{E}[\mathcal{K}_1^2(x)]}{\mathbb{E}^2[\mathcal{K}_1(x)]} \mathbb{E} \left[ \frac{\mathbb{B}_1^2}{\mathbb{E}[\mathcal{K}_1^2(x)]} \right] - \mathbb{E}^2 \left[ \frac{\mathbb{B}_1}{\mathbb{E}[\mathcal{K}_1(x)]} \right] \right). \quad (7.24)
\end{aligned}$$

Let us introduce the notation, for $i = 1, 2$

$$\mathcal{M}_i(x) = \mathcal{E}_i(v_p(u, x); X_1) - \mathcal{E}_i(\theta(p; x); x) \text{ and } \mathcal{N}_i(x) = L_i^{X_1}(v_p(u, x)) - L_i^x(\theta(p; x)).$$

Under the assumption (CO2) and (NA3), it follows immediately from the triangle inequality that, for $i \in \{1, 2\}$,

$$\begin{aligned}
d(x, X_1) \le h \left| \mathcal{M}_i(x) \right| &\le C_i h^{k_i} + u (n \phi_x(h))^{-\frac{1}{2}} \sigma_p(x) \mathcal{E}_i'(\theta(p; x); x), \\
d(x, X_1) \le h \left| \mathcal{N}_i(x) \right| &\le C_i h^{r_i} + u (n \phi_x(h))^{-\frac{1}{2}} \sigma_p(x) L_i'(\theta(p; x)).
\end{aligned}$$

By using double conditioning with respect to $X$ and $T$, we obtain

$$\begin{aligned}
\mathbb{E}\left[\frac{\mathbb{B}_1}{\mathbb{E}[\mathcal{K}_1(x)]}\right] &= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}\left[\mathcal{K}_1(x) \delta_1((\overline{\mathbf{G}}_n(Y_1))^{-1}) \left(\frac{p}{1-p} \psi_2(Y_1 - v_p(u, x)) + \psi_1(Y_1 - v_p(u, x))\right)\right] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}\left[\mathcal{K}_1(x)(\overline{\mathbf{G}}_n(Y_1))^{-1}(\overline{\mathbf{G}}_n(Y_1)) \left(\frac{p}{1-p} \mathcal{M}_2(x) - \mathcal{M}_1(x)\right)\right] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}\left[\mathcal{K}_1(x)\mathbb{E}\left[(\overline{\mathbf{G}}_n(Y_1))^{-1}(\overline{\mathbf{G}}_n(Y_1)) \mid T_1\right] \left(\frac{p}{1-p} \mathcal{M}_2(x) - \mathcal{M}_1(x)\right)\right] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}\left[\mathcal{K}_1(x)(\overline{\mathbf{G}}_n(T_1))^{-1}(\overline{\mathbf{G}}_n(T_1)) \left(\frac{p}{1-p} \mathcal{M}_2(x) - \mathcal{M}_1(x)\right)\right] \\
&= \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} \mathbb{E}\left[\mathcal{K}_1(x)\left(\frac{\overline{\mathbf{G}}_n(T_1) - \overline{\mathbf{G}}_n(T_1)}{\overline{\mathbf{G}}_n(T_1)} + 1\right)\left(\frac{p}{1-p} \mathcal{M}_2(x) - \mathcal{M}_1(x)\right)\right] \\
&\le \sup_{t \le \tau_F} \frac{|\overline{\mathbf{G}}_n(t) - \overline{\mathbf{G}}_n(t)|}{\overline{\mathbf{G}}_n(\tau_F)} o(1) + o(1) = o(1). \qquad (7.25)
\end{aligned}$$
By the same previous arguments used to treat the second term of 7.24, and by combining the assumptions (CO5) and (NA1)-(NA2), we have

$$
\begin{align}
\mathbb{E} \left[ \frac{\mathbb{B}_1^2}{\mathbb{E}[\mathcal{K}_1^2(x)]} \right] &= \mathbb{E} \left[ \frac{\mathcal{K}_1^2(x) \delta_1 ((\bar{\mathbf{G}}_n(Y_1))^{-2}) \left( \frac{p}{1-p} \psi_2^2(Y_1 - v_p(u, x)) + \psi_1^2(Y_1 - v_p(u, x)) \right)}{\mathbb{E}[\mathcal{K}_1^2(x)]} \right] \nonumber \\
&= \mathbb{E} \left[ \frac{\mathcal{K}_1^2(x) ((\bar{\mathbf{G}}_n(T_1))^{-2} \bar{\mathbf{G}}(T_1)) \left\{ \left( \frac{p}{1-p} \right)^2 \mathcal{N}_2(x) + \mathcal{N}_1(x) \right\}}{\mathbb{E}[\mathcal{K}_1^2(x)]} \right] + \gamma_p (\boldsymbol{\theta}(p; x); x) \nonumber \\
&\le \sup_{t \le \tau_F} \frac{|\bar{\mathbf{G}}_n(t) - \bar{\mathbf{G}}(t)|}{\bar{\mathbf{G}}_n^2(\tau_F)} o(1) + o(1) + \gamma_p (\boldsymbol{\theta}(p; x); x) = o(1) + \gamma_p (\boldsymbol{\theta}(p; x); x). \tag{7.26}
\end{align}
$$

We evaluate the term $\frac{\mathbb{E}[\mathcal{K}_1^2(x)]}{\mathbb{E}^2[\mathcal{K}_1(x)]}$ in the right side of (7.24). Using similar arguments as in Ferraty et al. (2007), we can show that

$$
\mathbb{E}[\mathcal{K}_1^j(x)] = \mathcal{K}_1^j(1)\phi_x(h) - \int_0^1 (\mathcal{K}_1^j)^{(\prime)}(s)\phi_x(sh)ds + o(\phi_x(h)) \text{ for } j \in \{1, 2\}. \quad (7.27)
$$

This when combined with (3.13) gives that, as $n \to \infty$,

$$
\frac{\phi_x(h) \mathbb{E}[\mathcal{K}_1^2(x)]}{\mathbb{E}^2[\mathcal{K}_1(x)]} \rightarrow \frac{\alpha_2(x)}{\alpha_1^2(x)}. \quad (7.28)
$$

Then, by combining (7.24), (7.25), (7.26), (7.27) and (7.28), we obtain the statement (7.22).
Now, we focus on the covariance term (7.23). To do that, we keep the notation of Lemma 3.4
and we use the same decomposition in 7.11. Then,

$$
\sum_{i \neq j} \mathrm{Cov}[\mu_i(x), \mu_j(x)] = \sum_{(i,j) \in S_1} \mathrm{Cov}[\mu_i(x), \mu_j(x)] + \sum_{(i,j) \in S_2} \mathrm{Cov}[\mu_j(x), \mu_j(x)].
$$

For the first part, we write

$$
\mathrm{Cov}[\mu_i(x), \mu_j(x)] \le \frac{C}{n\phi_x^2(h)} \left| \mathrm{E} \left[ \mathcal{K}_i(x) \mathcal{K}_j(x) \frac{\delta_i \delta_j}{\bar{\mathbf{G}}_n(Y_i) \bar{\mathbf{G}}_n(Y_j)} \right] \right| \\
+ \left| \mathrm{E} \left[ \mathcal{K}_i(x) \frac{\delta_i}{\bar{\mathbf{G}}_n(Y_i)} \right] \mathrm{E} \left[ \mathcal{K}_j(x) \frac{\delta_j}{\bar{\mathbf{G}}_n(Y_j)} \right] \right|.
$$

By the same argument in 7.25, we obtain

$$
\mathbb{E}^2[\mathbb{B}_1(x)] = 0(\phi_x^2(h)),
$$

and for $t \le \tau_G$

$$
\begin{align*}
\mathbb{E} [\mathbb{B}_i(x) \mathbb{B}_j(x)] &\le \mathbb{E} \left[ \mathcal{K}_i(x) \mathcal{K}_j(x) \frac{\delta_i \delta_j}{\bar{\mathbf{G}}_n(Y_i) \bar{\mathbf{G}}_n(Y_j)} \right] \\
&\le \mathbb{E} \left( \mathcal{K}_i(x) \mathcal{K}_j(x) \mathbb{E} \left[ \frac{\delta_i \delta_j}{\bar{\mathbf{G}}_n(Y_i) \bar{\mathbf{G}}_n(Y_j)} \mid T_i, T_j \right] \right) \\
&\le \frac{1}{\bar{\mathbf{G}}_n(T_i) \bar{\mathbf{G}}_n(T_j)} \mathbb{E} [\delta_i \delta_j \mid T_i, T_j]
\end{align*}
$$
$$ \leq \frac{C v_n}{n \phi_x^2(h)} \left( \sup_{t \leq \tau_G} \left\{ \frac{\overline{\mathbf{G}}^2(t)}{\overline{\mathbf{G}}_n^2(\tau_G)} \right\} \varphi_x(h) + \phi_x^2(h) \right), $$

then

$$ \sum_{S_1} \mathrm{Cov}[\mu_i(x), \mu_j(x)] \leq C \frac{v_n \chi_x(h)}{n \phi_x^2(h)}. $$

Now, we use Davydov-Rio's inequality to show that

$$ |\mathrm{Cov}(\mu_i(x), \mu_j(x))| \leq C \frac{1}{n^2 \phi_x^{(2m'-2)/m'}(h)} \sum_{(i,j) \in E_2} \alpha^{(m'-2)/m'}(|i-j|). $$

Hence, we have

$$ \sum_{S_2} |\mathrm{Cov}(\mu_i(x), \mu_j(x))| \leq C \frac{1}{\phi_x^{(2m'-2)/m'}(h)} v_n^{-a(m'-2)/m'}. $$

Choose

$$ v_n = \left( \frac{\chi_x(h)}{n \phi_x^{m'/2}(h)} \right)^{-\frac{1}{a((m'-2)/m') + 1}}, $$

we to prove that

$$ \sum_{i=1}^{n} \sum_{i \neq j} \mathrm{Cov}(\mu_i(x), \mu_j(x)) \leq \frac{C}{n \phi_x^2(h)} \left( \chi_x^{\frac{a((m'-2)/m')}{a((m'-2)/m') + 1}}(h) n^{\frac{1}{a((m'-2)/m') + 1}} \phi_x^{\frac{m'/2}{a((m'-2)/m') + 1}}(h) \right). $$

Under the second part of the condition (NA4), we get

$$ \begin{aligned} \sum_{i \neq j} \mathrm{Cov}[\mu_i(x), \mu_j(x)] &\leq C \frac{1}{n \phi_x(h)} \left[ n^{-\frac{1+a((m'-2)/m')\delta}{a((m'-2)/m') + 1}} \phi_x^{m'/2 - 2/m'}(h) \right] \\ &\leq C \frac{1}{n \phi_x(h)} n^{-\frac{\delta}{a((m'-2)/m') + 1}}, \end{aligned} $$

which complete the proof of (7.23). Now, to prove the asymptotic normality (7.5), we can apply the central limit theorem (Theorem 1.2) in Merlevède and Peligrad (2000), for

$$ Z_n = \frac{1}{\sqrt{n\sigma}} \sum_{i=1}^{n} Z_i \quad \text{and} \quad Z_i = \frac{1}{\sqrt{n}} \mu_i, $$

with

$$ \mu_i(x) = \frac{1}{n \mathbb{E}[K_1(x)]} (\mathbb{B}_i(x) - \mathbb{E}[\mathbb{B}_i(x)]), $$

$$ \mathbb{B}_i(x) = K_i(x) \delta_i (\overline{\mathbf{G}}_n(Y_i))^{-1} \left( \frac{p}{1-p} \psi_2(Y_i - v_p(u, x)) + \psi_1(Y_i - v_p(u, x)) \right), $$

$$ \sigma = \sqrt{\mathrm{Var}\left[\sum_{i=1}^{n} Z_i\right]} = \frac{1}{n} \sqrt{\frac{\alpha_2(x) \gamma_p(\theta(p; x); x)}{\phi_x(h) \alpha_1^2(x)}}. $$
The required condition of this central limit theorem is

$$ \lim_{n \to \infty} \sum_{i=1}^{n} \int_{0}^{\alpha(i)} [Q_{|Z_1|}(u)]^2 du < \infty, \quad (7.29) $$

where $Q_{|Z_1|}$ is the “upper tail” quantile function defined by

$$ Q_{|Z_1|}(u) = \inf\{t \ge 0 : \mathbb{P}(|Z_1| > t) \le u\}. $$

First, note that by Markov’s inequality, we show that

$$ Q_{|Z_1|}(u) \le \left( \frac{\mathbb{E}[|Z_1|^{m'}]}{u} \right)^{1/m'}, $$

with

$$ \mathbb{E}|Z_1|^{m'} = \frac{1}{n^{3/2}} \mathbb{E} \left| \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} (\mathbb{B}_1(x) - \mathbb{E}\mathbb{B}_1(x)) \right|^{m'}. $$

To calculate the expectation of the variable $|\mathbb{B}_1|^l$, and by (CO3) we notice that

$$ \begin{aligned} & \mathbb{E} \left[ \left| \frac{p}{1-p} \psi_2(Y_1 - v_p(u, x)) + \psi_1(Y_1 - v_p(u, x)) \right|^{m'} \middle| X_1 \right] \\ &= \mathbb{E} \left[ \left| \sum_{l=0}^{m'} C_l^{m'} \left( \frac{p}{1-p} \right)^l \psi_2^l(Y_1 - v_p(u, x)) + \psi_1^{m'-l}(Y_1 - v_p(u, x)) \right|^{m'} \middle| X_1 \right] \\ &\le \sum_{l=0}^{m'} C_l^{m'} \left( \frac{p}{1-p} \right)^l \mathbb{E} \left[ |\psi_2(Y_1 - v_p(u, x))|^l \middle| X_1 \right] + \mathbb{E} \left[ |\psi_1(Y_1 - v_p(u, x))|^{m'-l} \middle| X_1 \right] \\ &\le C. \end{aligned} $$

Then, under the condition (CO3), we have

$$ \begin{aligned} \mathbb{E}|\mathbb{B}_1|^l &\le C \mathbb{E} (\mathcal{K}_1^l(x) \delta_1 (\bar{\mathbf{G}}_n(Y_1))^{-l}) \\ &\le C \mathbb{E} (\mathcal{K}_1^l(x) (\bar{\mathbf{G}}_n(T_1))^{-l} \mathbb{E}[\delta_1 | T_1]) \\ &\le \frac{C}{\bar{\mathbf{G}}_n^l(\tau_G)} \sup_{t \le \tau_G} \{\bar{\mathbf{G}}(t)\} \mathbb{E}[\mathcal{K}_1^l(x)] \\ &\le C \phi_x(h). \end{aligned} $$

Hence, we have

$$ \begin{aligned} \mathbb{E} \left| \frac{1}{\mathbb{E}[\mathcal{K}_1(x)]} (\mathbb{B}_1(x) - \mathbb{E}\mathbb{B}_1(x)) \right|^{m'} &\le \frac{1}{\mathbb{E}^{m'}[\mathcal{K}_1(x)]} \mathbb{E} \left| \sum_{l=0}^{m'} C_l^{m'} \mathbb{B}_1^l(x) \mathbb{E}^{m'-l}(\mathbb{B}_1(x)) (-1)^{m'-l} \right| \\ &\le \frac{1}{\mathbb{E}^{m'}[\mathcal{K}_1(x)]} \sum_{l=0}^{m'} C_l^{m'} \mathbb{E}|\mathbb{B}_1(x)|^l \mathbb{E}|\mathbb{E}\mathbb{B}_1(x)|^{m'-l} \\ &\le C \max_{0 \le l \le m'} (\phi_x(h))^{-l+1} \le C (\phi_x(h))^{-m'+1}. \end{aligned} $$
Furthermore, we have

$$ \lim_{n \to \infty} \sum_{i=1}^{n} \int_{0}^{\alpha(i)} [Q_{\Lambda_1}(u)]^2 du \le C \lim_{n \to \infty} \frac{\phi_x^{2/m'}(h)}{n^3 \phi_x^2(h)} \sum_{i=1}^{n} i^{-a(2/m'-1)} < \infty. $$

Finally, this condition completes the proof of the asymptotic normality (7.5) and therefore that of the Lemma 7.3. □

**Acknowledgements** The authors would like to thank the Editor-in-Chief, an Associate-Editor, and the referees for their constructive remarks, which resulted in a substantial improvement of the work's original form and a more sharply focused presentation.

**Funding** Open access funding provided by Université de Technologie de Compiègne. This research project was funded by the Deanship of Scientific Research at King Khalid University through the Research Groups Program under grant number R.G.P./128/46.

**Data Availability** Not applicable.

## Declarations

**Conflicts of Interest** The authors declare no conflict of interest.

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