Research article Special Issues

The Burden of Grandparenting among Chinese older adults in the Greater Chicago area—The PINE Study

  • Grandparent caregiving responsibilities influence feelings of burden in many older adults. Though grandparenting is traditionally seen as a rewarding experience for Chinese older adults, some research has pointed to the possibility of grandparenting burden in Chinese older adults in the U.S. However, there is a paucity of research concerning prevalence of grandparenting burden in Chinese older adults. This study aims to provide an overall estimate on grandparenting burden and examine its correlations with socio-demographic characteristics, self-reported health data, and time spent caring for grandchildren. Data was collected through the Population Study of Chinese Elderly in Chicago (PINE) study. This community-based participatory research study surveyed a total of 3,159 Chinese older adults aged 60 and above, 2,146 of whom have grandchildren. We used four questions on a Likert-scale to determine levels of grandparenting burden. Our study found 22% of our participants who are grandparents experience grandparenting burden. Younger age (r = 0.12), living with four or more people (r = 0.09), lower overall health status (r = 0.09), lower quality of life (r = 0.09), and more time spent caring for grandchildren (r = 0.37) were correlated with grandparenting burden. Our findings show that grandparenting is not a necessarily rewarding experience for Chinese older adults, and that certain subsets of the population are more likely to experience grandparenting burden. Future longitudinal research should be conducted to determine causality as well the psychological, physical, and social effects of grandparenting burden.

    Citation: Xinqi Dong, E-Shien Chang, Stephanie Bergren BA. The Burden of Grandparenting among Chinese older adults in the Greater Chicago area—The PINE Study[J]. AIMS Medical Science, 2014, 1(2): 125-140. doi: 10.3934/medsci.2014.2.125

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  • Grandparent caregiving responsibilities influence feelings of burden in many older adults. Though grandparenting is traditionally seen as a rewarding experience for Chinese older adults, some research has pointed to the possibility of grandparenting burden in Chinese older adults in the U.S. However, there is a paucity of research concerning prevalence of grandparenting burden in Chinese older adults. This study aims to provide an overall estimate on grandparenting burden and examine its correlations with socio-demographic characteristics, self-reported health data, and time spent caring for grandchildren. Data was collected through the Population Study of Chinese Elderly in Chicago (PINE) study. This community-based participatory research study surveyed a total of 3,159 Chinese older adults aged 60 and above, 2,146 of whom have grandchildren. We used four questions on a Likert-scale to determine levels of grandparenting burden. Our study found 22% of our participants who are grandparents experience grandparenting burden. Younger age (r = 0.12), living with four or more people (r = 0.09), lower overall health status (r = 0.09), lower quality of life (r = 0.09), and more time spent caring for grandchildren (r = 0.37) were correlated with grandparenting burden. Our findings show that grandparenting is not a necessarily rewarding experience for Chinese older adults, and that certain subsets of the population are more likely to experience grandparenting burden. Future longitudinal research should be conducted to determine causality as well the psychological, physical, and social effects of grandparenting burden.


    Regression is the most frequently employed technique in nonparametric statistics to examine the association between two variables $ X $ and $ Y $. In this context, $ Y $ represents the response variable, while $ X $ is a random vector of predictors (covariates) that can assume values in the real number space $ \mathbb{R} $. The regression function at a point $ x\in\mathbb{R} $ is the conditional expectation of $ Y $ given $ X = x $, denoted as

    $ r(x): = \mathbb{E}(Y|X = x). $

    Various techniques can be employed to estimate a regression function, including kernel estimators, regression spline methods, and others. Nevertheless, these methods lack robustness as they are highly susceptible to outliers. Given that outliers are commonly observed in various fields, such as finance, it is essential to handle outliers properly to emphasize a dataset's unique features. Robust regression is a statistical technique used to address the issue of lack of robustness in regression models. It ensures that the model remains stable and resistant to the influence of outliers.

    Robust regression holds significant importance within the realm of statistics. It is employed to overcome certain constraints of non-robust regression, specifically when the data exhibit heteroscedasticity or include outliers. The earliest significant outcome in this field can be traced back to Huber's work in [1]. The regression estimation method mentioned has been extensively researched. For empirical data, notable studies include Robinson [2], Collomb and H$ \ddot{a} $rdle [3], Boente and Fraiman [4,5], and Fan et al. [6] for earlier findings. Recent advancements and references can be found in Laib and Ould-Saïd [7] and Boente and Rodriguez [8]. Traditional kernel estimators often exhibit significant bias near boundaries because the kernel's support can extend beyond them, resulting in inaccurate estimates. Being supported on the entire interval, Bernstein estimators do not suffer from this boundary bias, leading to more accurate estimations near the edges.

    The Bernstein polynomial is widely acknowledged as a valuable tool for interpolating functions on a closed interval, rendering it suitable for approximating density functions within that interval.

    The use of Bernstein polynomials as density estimators for variables with finite support has been proposed in several articles. Vitale [9] first introduced this concept, followed by Petrone [10,11]. Further studies on this topic were conducted by Babu, et al. [12], Petrone and Wassermann[13], and Kakizawa [14].

    Recently, Ouimet [15] studied some asymptotic properties of Bernstein cumulative distribution function and density estimators on the d-dimensional simplex and studied their asymptotic normality and uniform strong consistency. Belalia et al. [16] introduced a two-stage Bernstein estimator for conditional distribution functions. Various other statistical topics related to the Bernstein estimator have been treated; for more references, see Ouimet [15]. Khardani [17] investigated various asymptotic properties (bias, variance, mean squared error, asymptotic normality, uniform strong consistency) for Bernstein estimators of quantiles and cumulative distribution functions when the variable of interest is subject to random right-censoring.

    It is essential to mention that several authors have devised Bernstein-based methodologies for addressing non-parametric function estimation problems. Priestley and Chao[18] first proposed the potential application of Bernstein polynomials for regression problems. Tenbusch [19], Brown and Chen [20], Choudhuri, Ghosal, and Roy [21], Chang, Hsiung, Wu, and Yang [22], Kakizawa [23], and Slaoui and Jmaei [24] have all conducted research on various non-parametric function estimation problems.

    In this paper, our contribution is to find asymptotic expressions for the bias, variance, and mean squared error (MSE) for the Bernstein robust regression function estimator defined in (2.4) and (2.3) and also prove their asymptotic normality and convergence. We deduce the asymptotically optimal bandwidth parameter $ m $ using the expression for the MSE as well. The results provided by our Bernstein approach for the robust regression function are better than those of the traditional kernel estimators. In future work, using some kernels, such as Dirichlet, Wishart, and inverse Gaussian kernels, and the robust function will be investigated in other spaces, such as the simplex, the space of positive definite matrices, and half-spaces, etc.

    The subsequent sections of the paper are structured in the following manner. In the next section, we will introduce our model. Section 3 presents notations, assumptions, and investigates various asymptotic properties of the proposed estimator. Section 4 presents a simulation study that evaluates the proposed approach's performance compared to the Bernstein-Nadaraya-Watson estimator and the Nadaraya-Watson estimator. Section 5 discusses a real data application, while the proofs of the results are provided in the Appendix.

    Let $ (X, Y), \left(X_{1}, Y_{1}\right), \ldots, \left(X_{n}, Y_{n}\right) $ be independent, identically distributed pairs of random variables with joint density function $ g(x, y) $, and let $ f $ denote the probability density of $ X $, which is supported on $ [0, 1] $. Let $ x $ be a fixed element of $ \mathbb{R} $, and let $ \rho $ a real-valued Borel function that satisfies specific regularity conditions outlined below. The robust method used to study the links between $ X $ and $ Y $ belongs to the class of M-estimates introduced by Huber [1]. The robust nonparametric parameter studied in this work, denoted by $ \theta_x $, is implicitly defined as the unique minimizer w.r.t. $ t $ of

    $ r(x,t):=E(ρ(Yt)|X=x),
    $
    (2.1)

    that is

    $ θx=argmintRr(x,t).
    $
    (2.2)

    This definition covers and includes many important nonparametric models, for example, $ \rho(t) = t^2 $ yields the non-robust regression, $ \rho(t) = |t| $ leads to the conditional median function $ m(x) = \text{med}(Y \mid X = x) $, and the $ \alpha- $th conditional quantile is obtained by setting $ \rho(t) = |t| + (2\alpha - 1)(t) $. We return to Stone [25] for other examples of the function $ \rho $.

    We utilize the techniques outlined in Vitale [9] and Leblanc [26,27] for distribution and density estimation. Additionally, we refer to the work of Slaoui [28] and Tenbusch [19,29] for non-robust regression. Our objective is to establish a Bernstein estimator for robust regression, defined as

    $ ˆθx=argmintRˆrn(x,t),
    $
    (2.3)

    with at a given point $ x \in[0, 1] $ such that $ f(x) \neq 0 $ and

    $ ˆrn(x,t)=ni=1ρ(Yit)mn1k=0I{kmn<Xik+1mn}Bk(mn1,x)ni=1mn1k=0I{kmn<Xik+1mn}Bk(mn1,x)=Nn(x,t)fn(x),
    $
    (2.4)

    where $ B_{k}(m, x) = \left(mk

    \right) x^{k}(1-x)^{m-k} $ is the Bernstein polynomial of order $ m $. This estimator can be viewed as a generalization of the estimator proposed in Slaoui and Jmaei [28], with

    $ Nn(x,t)=mnnni=1ρ(Yit)mn1k=0I{kmn<Xik+1mn}Bk(mn1,x),
    $

    where $ f_n $ is Vitale's estimator of the density $ f $ defined, for all $ x \in[0, 1] $, by

    $ fn(x)=mnnni=1mn1k=0I{kmn<Xk+1mn}Bk(mn1,x)=mnmn1k=0{Fn(k+1mn)Fn(kmn)}Bk(mn1,x),
    $
    (2.5)

    with $ F_n $, the empirical distribution function of the variable $ X $.

    This paper will use the following notations:

    $ ψ(x)=(4πx(1x))1/2,Δ1(x)=12[(12x)f(x)+x(1x)f(x)],Δ2(x)=12{(12x)(rx(x,t)f(x)+f(x)r(x,t))+x(1x)(2f(x)rx(x,t)+f(x)2rx2(x,t)+f(x)r(x,t))},Δ(x)=12{x(1x)2rx2(x,t)+[(12x)+2x(1x)f(x)f(x)]rx(x,t)},δ1=10Δ2(x)dx,δ2=10Var[ρ(Yt)X=x]f(x)ψ(x)dx.
    $

    Moreover, we denote by $ o $ the pointwise bound in $ x $ (i.e., the error is not uniform in $ x\in [0, 1] $).

    Remark 2.1. Robust regression is advantageous in real data settings where outliers, non-normal errors, or heteroscedasticity are present, making it a more flexible and resilient choice.

    To state our results, we will need to gather some assumptions to make reading our results easier. In what follows, we will assume that the following assumptions hold:

    Throughout the paper, $ C_{1}, C_{2}, C_{3} $ represent positive constants, while $ C $ denotes a generic constant independent of $ n $. Let $ I_{0}: = \{x \in [0, 1] \; : \; f(x) > 0\} $ and $ S $ be a compact subset of $ I_{0} $.

    H1: $ m_{n}\geq2 $, $ m_{n}\underset{n \rightarrow +\infty}\longrightarrow\infty $ and $ m_{n} / n \underset{n \rightarrow +\infty}\longrightarrow0. $

    H2: $ g(s, t) $ is twice continuously differentiable with respect to $ s $.

    H3: For $ q \in\{0, 1, 2\}, s \mapsto \int_{\mathbb{R}} t^q g(s, t) d t $ is a bounded function continuous at $ s = x $.

    H4: For $ q > 2, s \mapsto \int_{\mathbb{R}}|t|^{-q} g(s, t) d t $ is a bounded function.

    H5: The function $ \rho(.) $ is a bounded, monotone, differentiable function. Its derivative is bounded.

    H6: The functions $ r $ and $ f $ are continuous and admit twice continuous and bounded derivatives such that $ | \frac{\partial r}{\partial x}(x, t)|\geq C > 0, $ $ \forall x \in \mathbb{R} $.

    H7: $ r(x, .) $ is of class $ \mathcal{C}^{1} $ on $ \left[ \theta_{x}-\tau, \theta_{x}+\tau\right] $ and satisfies $ \inf_{\left[ \theta_{x}-\tau, \theta_{x}+\tau\right]}\left| \frac{\partial r}{\partial t}(x, .) \right| > C_{3} $ and uniformly continuous.

    The assumptions we make are typical for this type of framework. Assumption (H1) is a technical requirement imposed to make proofs more concise. Assumptions (H2)–(H4) are necessary conditions for the estimation of the regression function in the couple $ (X, Y) $, as outlined in the works of Nadaraya [30], Watson [31], and Slaoui and Jmaei [28]. These assumptions pertain to the regularity of the density function. The condition (H5) controls the robustness properties of our model. It maintains the same conditions on the function $ \rho^{\prime} $ as those provided by Collomb and Härdle [3] and Boente and Rodriguez [8] in the multivariate case. Assumptions (H6) and (H7) deal with some regularity of the function $ r(., .) $. Note that condition (H6) is used to get the asymptotic normality of our estimator, and condition (H7) is somewhat less restrictive compared to that presented in the literature (see Boente and Fraiman [32], L. Aït Hennani, M.Lemdani, and E. Ould Saïd [33], Attouch et al.[34,35]), needed for the consistency result.

    Proposition 3.1. Under Assumptions (H1)–(H5), and for $ x\in \left[0, 1 \right] $ such that $ f(x) > 0 $, we have

    $ E[ˆrn(x,t)]r(x,t)=Δ(x)m1n+o(m1n),
    $
    (3.1)
    $ Var[ˆrn(x,t)]={m1/2nnE[(ρ(Yt))2X=x]f(x)ψ(x)+ox(m3/2nn)forx(0,1),mnnE[(ρ(Yt))2X=x]f(x)+ox(mnn)forx=0,1,
    $
    (3.2)
    $ MSE[ˆrn(x,t)]={Δ2(x)m2n+m1/2nnVar(ρ(Yt)X=x)f(x)ψ(x)+o(m2n)+ox(m1/2nn)ifx(0,1),Δ2(x)m2n+mnVar(ρ(Yt)X=x)f(x)+o(m2n)+ox(mnn)ifx=0,1.
    $
    (3.3)

    To minimize the $ MSE $ of $ \widehat{r}_{n} $, for $ x \in[0, 1] $ such that $ f(x) > 0 $, the order $ m_{n} $ must be equal to

    $ m_{o p t} = {[4Δ2(x)f(x)Var(ρ(Yt)X=x)ψ(x)]2/5n2/5ifx(0,1),[2Δ2(x)f(x)Var(ρ(Yt)X=x)]1/3n1/3ifx=0,1.
    $

    Then,

    $ M S E\left[\widehat{r}_{n, m_{o p t}}(x,t)\right] = {5(Δ(x))2/5(Var(ρ(Yt)X=x)ψ(x))4/5(4f(x))4/5n4/5+o(n4/5)ifx(0,1),3(Δ(x)Var(ρ(Yt)X=x))2/3(2f(x))2/3n2/3+o(n2/3)ifx=0,1.
    $

    Theorem 3.1. Under conditions of Proposition 3.1, we have

    $ \widehat{\theta}_{x}\underset{n \rightarrow +\infty}{\stackrel{\mathcal{P}}{\longrightarrow}}\theta_{x}. $

    Proposition 3.2. Let Assumptions (H1)–(H7) hold.

    1) For $ x \in(0, 1) $, we have:

    i) If $ n m_{n}^{-5 / 2} \underset{n \rightarrow +\infty}{\longrightarrow} c $ for some constant $ c \geq 0 $, then

    $ n1/2m1/4n(ˆrn(x,t)r(x,t))Dn+N(cΔ(x),Var(ρ(Yt)X=x)f(x)ψ(x)).
    $
    (3.4)

    ii) If $ n m_{n}^{-5 / 2} \underset{n \rightarrow +\infty}{\longrightarrow} \infty $, then

    $ mn(ˆrn(x,t)r(x,t))Pn+Δ(x).
    $
    (3.5)

    2) For $ x = \{0, 1\} $, we have:

    i) If $ n m_{n}^{-3} \underset{n \rightarrow +\infty}{\longrightarrow} c $ for some constant $ c \geq 0 $, then

    $ nm(ˆrn(x,t)r(x,t))Dn+N(cΔ(x),Var(ρ(Yt)X=x)f(x)).
    $
    (3.6)

    ii) If $ n m_{n}^{-3} \underset{n \rightarrow +\infty}{\longrightarrow} \infty $, then

    $ mn(ˆrn(x,t)r(x,t))Pn+Δ(x),
    $
    (3.7)

    where $ \underset{n \rightarrow +\infty}{\stackrel{\mathcal{D}}{\rightarrow }} $ denotes the convergence in distribution, $ \mathcal{N} $ the Gaussian distribution, and $ \underset{n \rightarrow +\infty}{\stackrel{\mathbb{P}}{\rightarrow }} $ the convergence in probability.

    Theorem 3.2. (The Mean Integrated Squared Error (MISE) of $ \widehat{r}_{n} $).

    Let Assumptions (H1)–(H7) hold. Then, we have

    $ MISE(ˆrn)=Λ1m2n+Λ2m1/2nn+o(m1/2nn)+o(m2n).
    $
    (3.8)

    Hence, the asymptotically optimal choice of $ m $ is

    $ m_{o p t} = \left[\frac{4 \Lambda_{1}}{\Lambda_{2}}\right]^{2 / 5} n^{2 / 5}, $

    for which we get

    $ \operatorname{MISE}\left(\widehat{r}_{n, m_{o p t}}\right) = \frac{5 \Lambda_{1}^{1 / 5} \Lambda_{2}^{4 / 5}}{4^{4 / 5}} n^{-4 / 5}+o\left(n^{-4 / 5}\right). $

    Theorem 3.3. Assume that (H1)–(H7) hold. If $ \Gamma(x, \theta_{x}) = \mathbb{E}\left[ \rho^{\prime} (Y-\theta_{x}) \rvert\, X = x\right]\neq 0 $, then $ \widehat{\theta}_{x} $ exists and is unique with great probability, and we have:

    i) when $ x \in(0, 1) $ and $ m_{n} $ is chosen such that $ n m_{n}^{-5 / 2} \rightarrow 0 $, then

    $ n^{1 / 2} m_{n}^{-1 / 4}(\widehat{\theta}_{x}-\theta_{x}) \xrightarrow{\mathcal{D}} \mathcal{N}\left(\frac{\sqrt{c} \Delta(x)}{\Gamma(x, \theta_{x})}, \sigma_{1}^{2}(x, \theta_{x})\right) , $

    ii) when $ x\in [0, 1] $ and $ m_{n} $ is chosen such that $ n m_{n}^{-3} \rightarrow 0 $, then

    $ \sqrt{\frac{n}{m_{n}}}(\widehat{\theta}_{x}-\theta_{x}) \xrightarrow{\mathcal{D}} \mathcal{N}\left(\frac{\sqrt{c} \Delta(x)}{\Gamma(x, \theta_{x})}, \sigma_{2}^{2}(x, \theta_{x})\right), $

    where

    $ \sigma_{1}^{2}(x, \theta_{x}) = \frac{\operatorname{Var}\left[\left. \rho(Y-\theta_{x}) \right\rvert\, X = x\right]}{f(x) \Gamma^{2}(x, \theta_{x})} \psi(x), \quad \sigma_{2}^{2}(x, \theta_{x}) = \frac{\operatorname{Var}\left[\left.\rho(Y-\theta_{x}) \right\rvert\, X = x\right]}{f(x) \Gamma^{2}(x, \theta_{x})}, $

    $ \underset{n \rightarrow +\infty}{\stackrel{\mathcal{D}}{\rightarrow }} $ denotes the convergence in distribution, and $ \mathcal{N} $ the Gaussian distribution.

    The following corollary directly follows from the previous theorem and provides the weak convergence rate of the estimator $ \widehat{\theta}_{x} $ for $ x\in [0, 1] $, where $ f(x) > 0 $. This is specifically for the case when $ m_{n} $ is chosen such that $ nm_{n}^{-5/2}\rightarrow0 $ for $ x\in (0, 1) $ and $ nm_{n}^{-3}\rightarrow0 $ for $ x\in [0, 1] $.

    Corollary 3.1. When $ x \in(0, 1) $ and $ m_{n} $ is chosen such that $ n m_{n}^{-5 / 2} \rightarrow 0 $, then

    $ n^{1 / 2} m_{n}^{-1 / 4}(\widehat{\theta}_{x}-\theta_{x}) \xrightarrow{\mathcal{D}} \mathcal{N}\left(0, \sigma_{1}^{2}(x, \theta_{x})\right). $

    When $ x\in [0, 1] $ and $ m_{n} $ is chosen such that $ n m_{n}^{-3} \rightarrow 0 $, then

    $ \sqrt{\frac{n}{m_{n}}}(\widehat{\theta_{x}}-\theta_{x}) \xrightarrow{\mathcal{D}} \mathcal{N}\left(0, \sigma_{2}^{2}(x, \theta_{x})\right), $

    where

    $ \sigma_{1}^{2}(x, \theta) = \frac{\operatorname{Var}\left[\left. \rho(Y-\theta_{x}) \right\rvert\, X = x\right]}{f(x) \Gamma^{2}(x, \theta_{x})} \psi(x), \quad \sigma_{2}^{2}(x, \theta_{x}) = \frac{\operatorname{Var}\left[\left.\rho(Y-\theta_{x}) \right\rvert\, X = x\right]}{f(x) \Gamma^{2}(x, \theta_{x})} . $

    This section is divided into two parts: the first shows our estimate's behavior for some particular conditional regression functions, and the second deals with asymptotic normality.

    Consider the regression model

    $ Y = r(X)+\varepsilon, $

    where $ \varepsilon \sim \mathcal{N}(0, 1) $.

    A simulation was conducted to compare the proposed estimators $ \widehat{\theta}_{x} $ (robust Bernstein polynomial estimator) with $ \widehat{r}^{BNW}_{n}(x) $ (Bernstein-Nadaraya-Watson estimator) introduced by Slaoui and Jmaei [28] and defined by

    $ ˆrBNWn(x)=ni=1Yimn1k=0I{kmn<Xik+1mn}Bk(mn1,x)ni=1mn1k=0I{kmn<Xik+1mn}Bk(mn1,x),
    $
    (4.1)

    where $ B_{k}(m, x) = \left(mk

    \right) x^{k}(1-x)^{m-k} $ is the Bernstein polynomial of order $ m $, and $ \widehat{r}^{NW}_{n}(x) $ (Nadaraya-Watson estimator) is defined, for $ x\in \mathbb{R} $ such that $ f(x)\neq 0 $, by

    $ ˆrNWn(x)=ni=1YiK(xXih)ni=1K(xXih),
    $
    (4.2)

    where $ K:\mathbb{R}\rightarrow \mathbb{R} $ is a nonnegative, continuous, bounded function satisfying $ \int_{\mathbb{R}} K(z)dz = 1, \int_{\mathbb{R}} z K(z)dz = 1 $ and $ \int_{\mathbb{R}} z^{2}K(z)dz < \infty $ and $ h = (h_{n}) $ is a sequence of positive real numbers that goes to zero.

    When using the estimator $ \widehat{r}^{NW}_{n}(x) $, we choose the Gaussian kernel $ K(x) = (2 \pi)^{-1 / 2} \exp \left(-x^2 / 2\right) $ and the bandwidth equal to $ \left(h_n\right) = m_n^{-1} $.

    We consider three sample sizes $ n = 20, n = 100 $, and $ n = 500 $, four regression functions

    $ Yi=2Xi+5+εilinear case,Yi=2X2i1+εiparabolic case,Yi=sin(32Xi)+εisine case,Yi=exp(2Xi3)+εiexponential case,
    $

    and three densities of $ X $: the truncated standard normal density $ \mathcal{N}_{[0, 1]}(0, 1) $ ($ X\in [0, 1] $), the exponential density $ Exp(2) $ ($ X\in [0, \infty) $), and the standard normal density $ \mathcal{N}(0, 1) $ ($ X\in (-\infty, \infty) $). It is also possible to use the transformations $ \widetilde{X} = \frac{X}{1 + X} $ or $ \widetilde{X} = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(X) $ to cover the cases of random variables $ X $ with support $ \mathbb{R}_+ $ and $ \mathbb{R} $, respectively. These transformations allow for the application of Bernstein polynomials to smooth the empirical distribution function.

    The simulation consists of four parts. In the first three parts, the estimators are compared by their average integrated squared error $ \overline{AISE} $. Every $ \overline{AISE} $ is calculated by a Monte-Carlo simulation with $ N = 1000 $ repetitions of sample size $ n $,

    $ \overline{AISE} = \frac{1}{N} \sum\limits_{k = 1}^N \operatorname{ISE}\left[\bar{r}_k\right], $

    where $ \bar{r}_k $ is the estimator ($ \widehat{\theta}_{x} $ or $ \widehat{r}^{BNW}_{n}(x) $ or $ \widehat{r}^{NW}_{n}(x) $) computed from the $ k^{\text {th }} $ sample, and

    $ \operatorname{ISE}\left[\bar{r}_k\right] = \displaystyle{\int}_0^1\{\bar{r}(x)-r(x)\}^2 d x. $

    According to Figures 14, it is evident that the robust Bernstein polynomial estimation converges when $ n $ is large. This is observed in all cases.

    Figure 1.  Prediction: linear case.
    Figure 2.  Prediction: parabolic case.
    Figure 3.  Prediction: sine case.
    Figure 4.  Prediction: exponential case.

    The $ \overline{AISE} $ of three estimators is graphed in Figure 5 for different parameter values ranging from 1 to 200. The estimators are evaluated for two sample sizes, $ n = 20 $ and $ n = 500 $. The outcomes are highly comparable when outlier values are not present. Nevertheless, the analysis of Tables 14 demonstrates that both the kernel estimator and the Bernstein-Nadaraya-Watson estimator exhibit significant sensitivity towards outlier values. This heightened sensitivity leads to substantial inaccuracies in predictions. In contrast, our robust Bernstein polynomial estimator consistently sustains its performance irrespective of the quantity of outlier values.

    Figure 5.  $ \overline{AISE} $ over the respective parameters in $ [1,200] $ for $ n = 20 $ and $ n = 500 $.
    Table 1.  $ \overline{AISE} $: linear case.
    Density of $ X $ Outlier rate $ n=20 $ $ n=100 $ $ n=500 $
    $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $
    (a)
    $ \mathcal{N}_{[0, 1]}(0, 1) $
    0.00% 0.37777 0.38362 0.37289 0.0386 0.04134 0.0333 0.01564 0.01684 0.00896
    0.05% 598.916 3.57632 690.548 678.998 2.20528 668.378 674.569 0.18818 692.737
    0.10% 3016.57 5.65957 3000.05 2620.97 3.16347 2593.3 2676.12 0.23244 2682.89
    0.25% 16083.1 14.182 15878.1 15896.3 6.29712 15930.6 16447.1 1.74161 16344.1
    (b)
    $ Exp(2) $
    0.00% 0.35574 0.35578 0.35517 0.05012 0.05283 0.03747 0.01611 0.01549 0.00794
    0.05% 748.855 4.2097 819.161 689.539 1.90398 692.123 683.571 0.21493 644.829
    0.10% 2408.65 5.93149 2284.35 2501.28 3.57741 2432.78 2681.96 0.31808 2586
    0.25% 16174.5 21.0217 16094.6 16228.2 6.62117 16834.3 17422.6 1.89746 17294.2
    (c)
    $ \mathcal{N}(0, 1) $
    0.00% 0.3345 0.33847 0.31945 0.05094 0.04832 0.03983 0.01667 0.01593 0.00886
    0.05% 770.807 4.51064 822.317 675.495 2.05097 665.198 698.8 0.15089 656.339
    0.10% 2746.52 7.79586 2559.23 2436.1 3.05173 2393.47 2497.94 0.24955 2503.83
    0.25% 19178.1 18.1898 18006 16413.5 8.0909 16893.9 17372.6 1.75941 17495.9

     | Show Table
    DownLoad: CSV
    Table 2.  $ \overline{AISE} $: parabolic case.
    Density of $ X $ Outlier rate $ n=20 $ $ n=100 $ $ n=500 $
    $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $
    (a)
    $ \mathcal{N}_{[0, 1]}(0, 1) $
    0.00% 1.48199 1.48019 1.47376 0.16874 0.25191 0.10863 0.04375 0.05576 0.02462
    0.05% 29.1125 2.49147 29.3273 14.8531 0.74622 16.0768 15.7507 1.66571 22.1559
    0.10% 64.0281 2.90274 73.1002 53.5638 0.91596 49.9706 98.0555 0.90805 67.9633
    0.25% 393.474 6.88132 326.212 1050.14 3.46544 700.888 1100.17 0.99655 751.459
    (b)
    $ Exp(2) $
    0.00% 1.39141 1.47957 1.33418 0.17829 0.22055 0.11882 0.05203 0.05178 0.02936
    0.05% 25.489 2.38298 28.3225 13.7623 0.54491 14.1639 31.6953 1.13268 11.5467
    0.10% 71.5908 2.58522 75.7357 50.8943 1.02112 58.2273 114.747 1.50033 54.9639
    0.25% 355.306 6.60588 289.937 867.431 2.26394 454.397 1327.96 0.47757 835.553
    (c)
    $ \mathcal{N}(0, 1) $
    0.00% 0.98856 1.05261 0.97223 0.16172 0.18081 0.09312 0.03957 0.04478 0.02101
    0.05% 25.528 2.25141 32.5343 22.6682 0.5839 15.2797 24.7481 1.48634 15.3258
    0.10% 61.7528 2.64854 85.8806 75.1123 1.22781 45.4987 111.539 1.81279 64.8843
    0.25% 469.185 9.35168 398.637 692.756 3.46427 642.386 1131.51 0.6867 526.238

     | Show Table
    DownLoad: CSV
    Table 3.  $ \overline{AISE} $: sine case.
    Density of $ X $ Outlier rate $ n=20 $ $ n=100 $ $ n=500 $
    $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $
    (a)
    $ \mathcal{N}_{[0, 1]}(0, 1) $
    0.00% 0.13301 0.12525 0.1154 0.01527 0.01522 0.01269 0.00414 0.00436 0.00316
    0.05% 19.3998 0.28717 20.5557 12.0525 0.12604 10.3004 9.47988 0.1339 8.91102
    0.10% 58.565 0.52241 49.1399 33.1922 0.23475 34.4606 38.2786 0.16123 43.7728
    0.25% 294.817 2.25567 177.939 266.142 0.54954 210.994 249.554 0.30488 273.319
    (b)
    $ Exp(2) $
    0.00% 0.14836 0.15054 0.12541 0.01324 0.0144 0.01196 0.0055 0.00545 0.00438
    0.05% 18.8191 0.2807 25.6982 10.7864 0.14311 9.74908 10.5176 0.18333 9.51805
    0.10% 55.1011 0.45925 48.4941 44.8719 0.18583 33.2084 40.9161 0.17936 39.3012
    0.25% 234.994 1.24542 189.692 251.89 0.60327 234.829 261.372 0.34955 285.042
    (c)
    $ \mathcal{N}(0, 1) $
    0.00% 0.13021 0.14029 0.12257 0.01506 0.01511 0.01375 0.00442 0.00442 0.00328
    0.05% 23.6259 0.28918 22.6131 12.1443 0.10116 10.529 9.82171 0.15066 9.71418
    0.10% 56.098 0.4286 55.5514 36.7151 0.22296 36.6241 35.7651 0.1501 40.4612
    0.25% 247.54 1.20312 237.361 224.049 0.50768 235.812 246.816 0.30212 276.141

     | Show Table
    DownLoad: CSV
    Table 4.  $ \overline{AISE} $: exponential case.
    Density of $ X $ Outlier rate $ n=20 $ $ n=100 $ $ n=500 $
    $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $ $ \widehat{r}^{BNW}_{n}(x) $ $ \widehat{\theta}_{x} $ $ \widehat{r}^{NW}_{n}(x) $
    (a)
    $ \mathcal{N}_{[0, 1]}(0, 1) $
    0.00% 0.74703 0.61209 0.56581 0.17137 0.17318 0.1214 0.10734 0.08411 0.01222
    0.05% 1.98561 1.35548 1.86444 1.56581 0.60637 1.98072 5.36892 0.08762 3.43111
    0.10% 5.07457 1.236 4.75581 7.88123 0.43664 7.38618 27.4856 0.11352 12.9594
    0.25% 34.2112 2.53764 30.5081 141.657 1.78744 166.463 366.283 0.32839 243.809
    (b)
    $ Exp(2) $
    0.00% 0.52866 0.5851 0.47474 0.10082 0.13134 0.05472 0.03985 0.07932 0.0121
    0.05% 1.79819 1.06047 1.9109 1.31466 0.45844 1.7421 4.25689 0.07832 4.11877
    0.10% 3.47787 1.33077 4.26777 10.9025 0.44565 11.0732 46.7737 0.09391 31.0859
    0.25% 66.6146 1.7682 51.3565 196.824 1.64135 113.157 351.074 0.30732 317.418
    (c)
    $ \mathcal{N}(0, 1) $
    0.00% 0.74933 0.71865 0.5849 0.11883 0.19364 0.11473 0.10082 0.11145 0.01105
    0.05% 1.20693 0.61275 1.51455 1.40167 0.15919 1.27418 7.52656 0.0597 3.79191
    0.10% 3.85005 1.01982 3.50089 14.9906 0.41717 10.9088 39.9271 0.11825 36.1256
    0.25% 26.8219 2.48908 25.5696 145.728 1.13974 123.627 422.568 0.33036 202.644

     | Show Table
    DownLoad: CSV

    The objective is to demonstrate the property of asymptotic normality in the context of the sine regression model. The equation is

    $ Yi=sin(32Xi)+εi.
    $

    Next, let $ r(x) $ be defined as the sine function with a coefficient of $ \frac{3}{2} $. The data provided is the same as in the previous subsection. The procedure consists of the following steps: We approximate the regression function $ r(x) $ using $ \widehat{\theta}_{x_0} $ and compute the normalized deviation between this approximation and the theoretical regression function (refer to Theorem 3.3) for $ x_0 = 0, \; 0.5\; \text{and}\; 1 $. Under this scheme, we generate $ N $ separate sets of $ n $ samples that are not influenced by each other. Next, we analyze the form of the estimated density (with normalized deviation) and compare it to the shape of the standard normal density in the context of the sine regression model. The following figures and table present the density of $ \widehat{\theta}_{x_0} $ as well as the $ p- $value by the Shapiro-Wilk normality test. We examine various values of $ n $, specifically $ n = 20 $, $ n = 100 $, and $ n = 500 $.

    Figures 68 and Table 5 demonstrate the advantageous characteristics of our asymptotic law compared to the standard normal distribution.

    Figure 6.  Illustration of the asymptotic normal distribution for $ x_0 = 0 $.
    Figure 7.  Illustration of the asymptotic normal distribution for $ x_0 = 0.5 $.
    Figure 8.  Illustration of the asymptotic normal distribution for $ x_0 = 1 $.
    Table 5.  $ p- $value by Shapiro-Wilk normality test.
    $ n=20 $ $ n=100 $ $ n=500 $
    $ x_0=0 $ 0.0814 0.0968 0.1728
    $ x_0=0.5 $ 0.5299 0.5734 0.6603
    $ x_0=1 $ 0.0611 0.0702 0.0970

     | Show Table
    DownLoad: CSV

    Air pollution significantly affects the lives of individuals in developed nations. The source of this issue is increased levels of smoke produced by industries or vehicles, prompting authorities to search for more efficient methods to regulate air quality in real-time. London is experiencing a significant problem with air pollution exceeding legal and World Health Organisation limits. An example of this is the incident in 2010 when air pollution caused various health problems in the city, leading to a financial cost of around £3.7 billion.

    This segment analyzes the mean daily levels of gases detected at the Marylebone Road monitoring station in London. The dataset includes the average daily measurements recorded throughout 2022 for five important variables: Ozone ($ O_3 $), Nitric Oxides ($ NO $), Nitrogen Dioxide ($ NO_2 $), Sulphur Dioxide ($ SO_2 $), and Particulate Matter ($ PM_{10} $). The main objective of our research is to determine the most practical forecasting models for air pollutant concentration. The data used in this analysis was obtained from the specified website: https://www.airqualityengland.co.uk/site/data?site_id = MY1.

    To ensure clarity, let us delineate the mathematical expression representing our prediction objective. Let us consider predicting the daily air pollutant concentration, represented by the variable $ Y $, for 365 days, denoted by $ X $. Formally, we assume that the output variable $ Y $ and the input variable $ X $ are connected by the following equation:

    $ Y_i = r\left(X_i\right)+\varepsilon_i \quad \text{for } i \in\{1, \ldots, n\}. $

    A dependable data-dependent rule for order selection is crucial when estimating an unknown regression function in any practical scenario. A widely used and effective method is cross-validation:

    $ C V(m) = \frac{1}{n} \sum\limits_{i = 1}^n \left(Y_i-\bar{r}_{-i}\left(X_i\right)\right)^2, $

    where $ \bar{r}_{-i} $ is the regression estimate without the data point $ \left(X_i, Y_i\right) $.

    In practice, choosing the right degree $ m $ for a Bernstein polynomial requires balancing between the complexity of the model and how well it fits the data. A useful method for this is cross-validation, where the dataset is divided into training and validation sets.

    Then, the smoothing parameter is chosen by minimizing

    $ C V(m) = \frac{1}{n} \sum\limits_{i = 1}^n \left(Y_i-\bar{r}_{-i}\left(X_i\right)\right)^2. $

    For convenience, we assume that the minimum of days is 1 and the maximum is 365 (the day data are such that $ \min _i\left(X_i\right) = 1 $ and $ \max _i\left(X_i\right) = 365 $). Finally, we used the cross-validation method to obtain the results in Figures 913 and Table 6.

    Figure 9.  Prediction: Ozone ($ O_3 $) case.
    Figure 10.  Prediction: Nitric Oxides ($ NO $) case.
    Figure 11.  Prediction: Nitrogen Dioxide ($ NO_2 $) case.
    Figure 12.  Prediction: Sulphur Dioxide ($ SO_2 $) case.
    Figure 13.  Prediction: Particulate Matter ($ PM_{10} $) case.
    Table 6.  $ m $ optimal for each case.
    Ozone Nitric Oxides Nitrogen Dioxide Sulphur Dioxide Particulate Matter
    $ \widehat{r}^{BNW}_{n}(x) $ 181 169 197 197 197
    $ \widehat{\theta}_{x} $ 121 149 101 173 181

     | Show Table
    DownLoad: CSV

    Based on the analysis of Figures 9 to 13, it is evident that the two estimators are nearly identical, except for the scenario depicted in Figure 10. In this case, non-robust estimator $ \widehat{r}^{BNW}_{n}(x) $ is found to be sensitive to outliers, which provides evidence of the efficiency of our estimator.

    Based on the information in Table 6, we can infer that the parameter $ m $ can be adjusted. It does not need to be equal to $ n $. Instead, we can choose a lower-degree polynomial to achieve a more favorable outcome.

    In this paper, we proposed a new robust regression estimator based on the Bernstein polynomials. Our contribution extends the work of Slaoui and Jmaei [28] to the case of robust regression. The asymptotic properties of this estimator were established. Afterward, we validated the effectiveness of the proposed method through a simulation study and applied it to real data on air pollution,

    We found that, in all three models, the average ISE of our robust regression estimator $ \widehat{\theta}_{x} $, defined in 2.4, was the smallest. We also noted that the robust regression provided better results than the non-robust method when outliers were present, in the sense that, even if the sample size increases, the average ISE decrease. To conclude, the use of the robust regression estimator with Bernstein polynomials successfully addressed the edge problem, yielding results comparable to those of non-robust and Nadaraya-Watson estimators in the absence of outliers.

    We believe our research provides a foundational step that can be further developed and expanded. It sets the stage for future work to extend our robust regression estimator using the Bernstein polynomial by considering the interest random variable to be truncated. We also plan to work on the robust regression estimation using Lagrange polynomials.

    Sihem Semmar: Conceptualization, data curation, formal analysis, investigation, methodology, software, validation, writing original draft, writing – review & editing; Omar Fetitah, Salah Khardani and Mohammad Kadi Attouch: Conceptualization, supervision, writing–review & editing; Mohammed Kadi Attouch and Ibrahim M. Almanjahie: Resources, validation, writing–review & editing. All authors have read and approved the final version of the manuscript for publication.

    The real data used in this application can be found at this link: https://www.airqualityengland.co.uk/site/data?site_id = MY1

    The authors thank and extend their appreciation to the funder of this work. This work was supported by the Deanship of Scientific Research and Graduate Studies at King Khalid University through the Large Research Groups Project under grant number R.G.P. 2/338/45.

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

    In this section, we present proofs for the results in Section 3. First, we recall a series of results, which are proven in Leblanc [26], linked to different sums of Bernstein polynomial, defined by

    $ S_{m_{n}}(x) = \sum\limits_{k = 0}^{m_{n}-1}B^{2}_{k}(m_{n},x). $

    These results are given in the following lemma.

    Lemma A.1. We have

    (i) $ 0 \leq S_{m_{n}}(x) \leq 1, \forall x \in[0, 1] $.

    (ii) $ S_{m_{n}}(x) = m^{-1 / 2}\left[\psi(x)+o_x(1)\right], \forall x \in(0, 1) $.

    (iii) $ S_{m_{n}}(0) = S_{m_{n}}(1) = 1 $.

    (iv) Let $ g $ be any continuous function on $ [0, 1] $. Then, $ m_{n}^{1 / 2} \int_0^1 g(x) S_{m_{n}(x)}dx = \int_0^1 g(x) \psi(x) d x+o(1) $.

    Proof. The proof of this lemma is in Leblanc [26] and Babu et al. [12].

    Lemma A.2.

    $ E[Nn(x,t)]N(x,t)=Δ2(x)m1n+o(m1n).
    $
    (A.1)

    Proof.

    $ E[Nn(x,t)]=mnE[ρ(Yt)mn1k=0I{kmn<Xk+1mn}Bk(mn1,x)]=mnmn1k=0k+1mnkmn(Rρ(yt)g(z,y)dy)dzBk(mn1,x)=mnmn1k=0(k+1mnkmnr(z,t)f(z)dz)Bk(mn1,x).
    $

    Using a Taylor expansion, we have

    $ r(z,t)f(z)=[r(x,t)+(zx)rz(x,t)+(zx)222rz2(x,t)+o((zx)2)]×[f(x)+(zx)f(x)+(zx)22f(x)+o((zx)2)]=r(x,t)f(x)+(zx)[rz(x,t)f(x)+r(x,t)f(x)]+(zx)22[2rz2(x,t)f(x)+f(x)r(x,t)+2rz(x,t)f(x)]+o((zx)2),
    $

    and since $ N(x, t) = r(x, t)f(x) $, we obtain

    $ E[Nn(x,t)]=r(x,t)f(x)mnmn1k=0(k+1mnkmn)Bk(mn1,x)+(rx(x,t)f(x)+f(x)r(x,t))mn2mn1k=0{(k+1mnx)2(kmnx)2}Bk(mn1,x)+(f(x)rx(x,t)+f(x)2rx2(x,t)+f(x)r(x,t))mn6mn1k=0{(k+1mnx)3(kmnx)3}Bk(mn1,x)=N(x,t)+(rx(x,t)f(x)+f(x)r(x,t))mn2mn1k=0m2n(2k+12mnx)Bk(mn1,x)+(2f(x)rx(x,t)+f(x)2rx2(x,t)+f(x)r(x,t))mn6mn1k=0m3n{(k+1mnx)2+(kmnx)2+(k+1mnx)(kmnx)}Bk(mn1,x)[1+o(1)]=N(x,t)+(rx(x,t)f(x)+f(x)r(x,t))m1n2{2T1,mn1(x)+(12x)T0,mn1(x)}+(2f(x)rx(x,t)+f(x)2rx2(x,t)+2fx2(x)r(x,t))m2n6mn1k=0{3(kmnx)2+3(kmnx)+1}Bk(mn1,x)[1+o(1)]=N(x,t)+(rx(x,t)f(x)+f(x)r(x,t))m1n2{2T1,mn1(x)+(12x)T0,mn1(x)}+(2f(x)rx(x,t)+f(x)2rx2(x,t)+f(x)r(x,t))m2n6{3T2,mn1(x)+3(12x)T1,mn1(x)+(x23x+1)T0,mn1(x)}[1+o(1)],
    $

    where $ T_{j, m_{n}-1}(x) $ are the central moments of the Binomial distribution of order $ j\in \mathbb{N} $, defined as

    $ T_{j, m_{n}-1}(x) = \sum\limits_{k = 0}^{m_{n}-1}(k-m_{n}x)^{j} B_{k}(m_{n}-1, x), \quad \forall j \in \mathbb{N} . $

    Note that it is easy to obtain

    $ T_{0, m_{n}-1}(x) = 1, \quad T_{1, m_{n}-1}(x) = 0 \quad T_{2, m_{n}-1}(x) = (m_{n}-1)x(1-x). $

    Then, we have

    $ E[Nn(x,t)]=N(x,t)+Δ2(x)m1n+o(m1n).
    $
    (A.2)

    Lemma A.3. We have

    $ \operatorname{ Var }\left[N_n(x,t)\right] = {m1/2nnE[(ρ(Yt))2X=x]f(x)ψ(x)+ox(m3/2nn)forx(0,1),mnnE[(ρ(Yt))2X=x]f(x)+ox(mnn)forx=0,1.
    $

    Proof. We have

    $ \operatorname{Var}\left[N_{n}(x,t)\right] = \mathbb{E}\left[N_{n}^{2}(x,t)\right]-\mathbb{E}^{2}\left[N_{n}(x,t)\right], $

    where

    $ N2n(x,t)=m2nn2ni=1(ρ(Yit))2(mn1k=0I{kmn<Xik+1mn}Bk(mn1,x))2+m2nn2mni,j=1,ijρ(Yit)ρ(Yjt)(mn1k=0I{kmn<Xik+1mn}Bk(mn1,x))(mn1k=0I{kmn<Xjk+1mn}Bk(mn1,x)).
    $

    So, we have

    $ E[N2n(x,t)]=m2nnE[(ρ(Yt))2(mn1k=0I{kmn<Xk+1mn}Bk(mn1,x))2]+m2nn(n1)n2E2[mn1k=0I{kmn<Xk+1mn}Bk(mn1,x)]=mnnE[(ρ(Yt))2(mn1k=0I{kmn<Xk+1mn}Bk(mn1,x))2]+(11n)E2[Nn(x,t)],
    $

    and

    $ Var[Nn(x,t)]=m2nnE[(ρ(Yt))2(mn1k=0I{kmn<Xk+1mn}Bk(mn1,x))2]1nE2[Nn(x,t)]=m2nnE[(ρ(Yt))2mn1k=0I{kmn<Xk+1mn}B2k(mn1,x)]1nE2[Nn(x,t)]=m2nnmn1k=0k+1mnkmn(R(ρ(yt))2g(z,y)dy)dzB2k(mn1,x)1nE2[Nn(x,t)]=m2nnmn1k=0(k+1mnkmnE[(ρ(Yt))2X=z]f(z)dz)B2k(mn1,x)1nE2[Nn(x,t)]=mnnE[(ρ(Yt))2X=x]f(x)Smn(x)1nE2[Nn(x,t)].
    $

    Using Lemma A.1 (ⅱ) and (ⅲ), we obtain

    $  Var [Nn(x,t)]={m1/2nnE[(ρ(Yt))2X=x]f(x)ψ(x)+ox(m3/2nn) for x(0,1),mnnE[(ρ(Yt))2X=x]f(x)+ox(mnn) for x=0,1.
    $
    (A.3)

    Lemma A.4.

    $ Cov(fn(x),Nn(x,t))={m1/2nnr(x,t)f(x)ψ(x)+ox(m1/2nn)forx(0,1),mnnr(x,t)f(x)+ox(mnn)forx=0,1.
    $
    (A.4)

    Proof. We have

    $ Cov(fn(x),Nn(x,t))=E[fn(x)Nn(x,t)]E[fn(x)]E[Nn(x,t)]=m2nnE[ρ(Yt)(mn1k=0I{kmn<Xk+1mn}Bk(mn1,x))2]+n(n1)m2nn2E2[ρ(Yt)mn1k=0I{kmn<Xk+1mn}Bk(mn1,x)]E[fn(x)]E[Nn(x,t)]=m2nnE[ρ(Yt)(mn1k=0I{kmn<Xk+1mn}Bk(mn1,x))2]1nE[fn(x)]E[Nn(x,t)]=m2nnmn1k=0k+1mnkmn(Rρ(yt)g(z,y)dy)dzB2k(mn1,x)1nE[fn(x)]E[Nn(x,t)]=mnnr(x,t)f(x)Sm(x)1nE[fn(x)]E[Nn(x,t)]
    $

    Using Lemma A.1 (ⅱ) and (ⅲ), we get

    $ Cov(fn(x),Nn(x))={m1/2nnr(x)f(x)ψ(x)+ox(m1/2nn) for x(0,1),mnnr(x)f(x)+ox(mnn) for x=0,1.
    $
    (A.5)

    To obtain the bias of $ \widehat{r}_{n}(x, t) $, we let $ h(x, y) = \frac{u}{v} $. Using a Taylor expansion, we have

    $ h(u,v)=h(u0,v0)+[uu0]hu(u0,v0)+[vv0]hv(u0,v0)+12{[uu0]22hu2(u0,v0)+[vv0]22qv2(u0,v0)}+2[uu0][vv0]2huv(u0,v0)+o((uu0,vv0)2).
    $

    Then, we have

    $ \frac{u}{v} = \frac{u_0}{v_0}+\frac{1}{v_0}\left(u-u_0\right)-\frac{u_0}{v_0^2}\left(v-v_0\right)+\frac{u_0}{v_0^3}\left(v-v_0\right)^2-\frac{1}{v_0^2}\left(u-u_0\right)\left(v-v_0\right)+o\left(\left(u-u_0\right)^2+\left(v-v_0\right)^2\right). $

    We set $ \left(u, v\right) = \left(N_{n}(x, t), f_{n}(x)\right) $ and $ (u_{0}, v_{0}) = (N(x, t), f(x)) $. Therefore, we infer that

    $ Nn(x,t)fn(x)=N(x,t)f(x)+1f(x)(Nn(x,t)N(x,t))N(x,t)f(x)2(fn(x)f(x)(x))+N(x,t)f(x)3(fn(x)f(x))21f(x)2(Nn(x,t)N(x,t))(fn(x)f(x)))+o((Nn(x,t)N(x,t))2+(fn(x)f(x))2).ˆrn(x,t)=r(x,t)+1f(x)(Nn(x,t)N(x,t))r(x,t)f(x)(fn(x)f(x))+r(x,t)f(x)2(fn(x)f(x))21f(x)2(Nn(x,t)N(x,t))(fn(x)f(x))+o((Nn(x,t)N(x,t))2+(fn(x)f(x))2).
    $

    Hence, we set $ \left(u, v\right) = \left(f_{n}(x), N_{n}(x, t)\right) $ and $ (u_{0}, v_{0}) = (f(x), N(x, t)) $ to obtain

    $ ˆrn(x,t)=r(x,t)r(x,t)f(x)(fn(x)f(x))+1f(x)(Nn(x,t)N(x,t))+r(x,t){f(x)}2(fn(x)f(x))21{f(x)}2(fn(x)f(x))(Nn(x,t)N(x,t))+o((fn(x)f(x))2+(fn(x)f(x))(Nn(x,t)N(x,t))).
    $

    Then,

    $ E[ˆrn(x,t)]=r(x,t)r(x,t)f(x)(E[fn(x)]f(x))+1f(x)(E[Nn(x,t)]N(x,t))+r(x,t){f(x)}2(E[fn(x)]f(x))21{f(x)}2E[(fn(x)f(x))(Nn(x,t)N(x,t))]+o(E[(fn(x)f(x))2]+E[(fn(x)f(x))(Nn(x,t)N(x,t))]).
    $

    Use Vitale's estimator $ f_{n} $, we get

    $ E[fn(x)]=f(x)+Δ1(x)mn+o(m1n),x[0,1]
    $
    (A.6)

    and

    $ Var[fn(x)]={m1/2nnf(x)ψ(x)+ox(m1/2nn) for x(0,1),mnnf(x)+ox(mnn) for x=0,1.
    $
    (A.7)

    To obtain (3.1) of Proposition 3.1, we use (A.6) and (A.2) to obtain

    $ E[ˆrn(x,t)]=r(x,t)+(1f(x)Δ2(x)r(x,t)f(x)Δ1(x))m1n+o(m1n)=r(x,t)+Δ(x)m1n+o(m1n),x[0,1].
    $

    Now for the variance of $ \widehat{r}_{n}(x, t) $, we have

    $ \operatorname{Var}\left(\widehat{r}_{n}(x,t)\right) = \operatorname{Var}\left( r(x,t)-\frac{r(x,t)}{f(x)}\left(f_{n}(x)-f(x)\right)+\frac{1}{f(x)}\left(N_{n}(x,t)-N(x,t)\right)\right) [1+o(1)], $

    which ensures that

    $ \operatorname{Var}\left(\widehat{r}_{n}(x,t)\right)\left\lbrace\frac{r^{2}(x,t)}{f^{2}(x)}\operatorname{Var}\left(f_{n}(x)\right)+\frac{1}{f^{2}(x)} \operatorname{Var}\left(N_{n}(x,t)\right)-2\frac{r(x,t)}{f^{2}(x)}\operatorname{Cov}\left(N_{n}(x,t), f_{n}(x)\right) \right\rbrace [1+o(1)]. $

    So, for $ x = (0, 1) $, we have $ f $,

    $ \operatorname{Var}\left[\widehat{r}_{n}(x,t)\right] = \frac{m_{n}^{1/2}}{n} \frac{\operatorname{Var}(\rho(Y-t) \mid X = x)}{f(x)}+o_{x}\left(\frac{m_{n}^{1/2}}{n}\right), $

    and, for $ x \in {0, 1} $, we have

    $ \operatorname{Var}\left[\widehat{r}_{n}(x,t)\right] = \frac{m_{n}}{n} \frac{\operatorname{Var}(\rho(Y-t) \mid X = x)}{f(x)}+o_{x}\left(\frac{m_{n}}{n}\right), $

    which gives the proof of Proposition 3.1.

    Without loss of generality we can suppose that $ \rho(Y-.) $ is increasing, with the decreasing case being obtained by considering $ -\rho(Y-.) $. As $ \rho(Y-.) $ is increasing, then for all $ \epsilon > 0 $,

    $ r(x,\theta_{x_{.}}+\epsilon)\leq r(x,\theta_{x})\leq r(x,\theta_{x}-\epsilon). $

    Proposition 3.1 shows that

    $ \widehat{r}(x,t) \xrightarrow{\mathbb{P}} r(x,t), $

    for all real $ t \in[\theta_{x}-\tau, \theta_{x}+\tau] $. As $ r(x, \theta_{x}) = 0 $, for sufficiently large $ n $ and for all $ \epsilon \leq \tau, $ this implies

    $ \widehat r(x,\theta_{x}+\epsilon)\leq 0\leq \widehat r(x,\theta_{x}-\epsilon) \quad \text{in probability}. $

    Since $ \widehat r(x, \widehat \theta_{x}) = 0 $, and by the continuity of $ \widehat r(x, $.$) $ on $ [\theta_{x}-\tau, \theta_{x}+\tau] $, we deduce that

    $ \theta_{x}-\epsilon \leq \hat{\theta_{x}} \leq \theta_{x}+\epsilon \quad \text{in probability}. $

    On the other hand, since $ \theta_{x} $ and $ \hat{\theta_{x}} $ are solutions of $ r(x, t) $ and $ \widehat{r}(x, t) $, respectively, then we have

    $ \widehat{r}(x,\hat{\theta_{x}}) = r(x, \theta_{x}) = 0 . $

    Under (H7), and by a Taylor expansion of $ r(x, $.$) $ of order one around $ \hat\theta_{x} $, we have

    $ \widehat{r}(x, \hat\theta_{x})-r(x, \hat\theta_{x}) = (\theta_{x}-\hat\theta_{x}) \frac{\partial r}{\partial t}\left(x, \xi_{n}\right), $

    where $ \xi_{n} $ is between $ \theta_{x} $ and $ \hat{\theta_{x}} $. Hence,

    $ |\theta_{x}-\hat{\theta_{x}}|\leq \frac{1}{|\inf\nolimits_{x \in S}\frac{\partial r}{\partial t}\left(x, \xi_{n}\right)|}\left|\hat{r}(x,\hat{\theta_{x}})-r(x,\hat{\theta_{x}})\right|, $

    which yields

    $ supxS|θx^θx|1C3supxS|ˆr(x,^θx)r(x,^θx)|1C3supxSsupt[θxτ,θx+τ]|ˆr(x,t)r(x,t)|,
    $

    and the rest of the proof is a sequence of Proposition 3.1.

    From (2.4), we adopt the decomposition stated as

    $ ˆrn(x,t)r(x,t)=1fn(x)[(Nn(x,t)N(x,t))r(x,t)(fn(x)f(x))]=1fn(x)[(Nn(x,t)E(Nn(x,t)))r(x,t)(fn(x)E(fn(x))))]+1fn(x)[(E(Nn(x,t))N(x,t))r(x,t)(E(fn(x))f(x))].
    $

    Lemma A.5. Under Assumptions (H1)–(H3), and for $ x \in[0, 1] $ such that $ f(x) > 0 $, we have

    $ fn(x)Pf(x).
    $
    (A.8)

    Proof. We have by the results of Lemmas A.2 and A.3, that

    $ \mathbb{E}\left(f_{n}(x)\right)-f(x) \rightarrow 0, $

    and

    $ \operatorname{Var}\left(f_{n}(x)\right) \rightarrow 0. $

    Hence,

    $ f_{n}(x)\xrightarrow{\mathbb{P}} f(x),\, \forall\,\, x \in(0,1). $

    Lemma A.6. Under Assumptions (H1)–(H4), and for $ x \in(0, 1) $ such that $ f(x) > 0 $, we have:

    i) if $ m_n $ is chosen such that $ n m_n^{-5 / 2} \rightarrow c $ for some constant $ c \geq 0 $, then

    $ n1/2m1/4nfn(x)[(E(Nn(x,t))N(x,t))r(x,t)(E(fn(x))f(x))]PcΔ(x),
    $
    (A.9)

    ii) if $ m_n $ is chosen such that $ n m_n^{-5 / 2} \rightarrow \infty $, then

    $ mnfn(x)[(E(Nn(x,t))N(x,t))r(x,t)(E(fn(x))f(x))]PΔ(x).
    $
    (A.10)

    Proof. By Lemmas A.2 and A.8, we have:

    i) if $ n m_n^{-5 / 2} \rightarrow c $ for some constant $ c \geq 0 $, then

    $ n1/2m1/4nfn(x)[(E(Nn(x,t))N(x,t))r(x,t)(E(fn(x))f(x))]=n1/2m5/4n(Δ1(x)r(x,t)Δ2(x)+o(1))fn(x)PcΔ(x),
    $

    ii) if $ n m_n^{-5 / 2} \rightarrow \infty $, then

    $ mnfn(x)[(E(Nn(x,t))N(x,t))r(x,t)(E(fn(x))f(x))]=(Δ1(x)r(x,t)Δ2(x)+o(1))fn(x)PΔ(x).
    $

    Lemma A.7. Under Assumptions (H1)–(H4), and for $ x \in(0, 1) $ such that $ f(x) > 0 $, we have

    $ n1/2m1/4n[(Nn(x,t)E(Nn(x,t)))r(x,t)(fn(x)E(fn(x)))]DN(0,Var(ρ(Yt)X=x)f(x)ψ(x)).
    $
    (A.11)

    Proof. We write

    $ n^{1/2} m_n^{-1 / 4}\left[\left(N_{n}(x,t)-\mathbb{E}\left(N_{n}(x,t)\right)\right)-r(x,t)\left(f_{n}(x)-\mathbb{E}\left(f_{n}(x)\right)\right)\right] = \sum\limits_{i = 1}^n\left(L_i(x)-\mathbb{E}\left(L_i(x)\right)\right), $

    where

    $ Li(x)=m3/4nn1/2(ρ(yit)r(x,t))mn1k=0I{kmn<Xik+1mn}Bk(mn1,x).
    $

    The proof of this lemma is based on the Lyapunov central limit theorem (FELLER, W.[36]) on $ L_i(x) $, i.e., it suffices to show, for some $ \delta > 0 $, that

    $ ni=1E[|Li(x)E[Li(x)]|2+δ](Var[ni=1Li(x)])(2+δ)/20.
    $
    (A.12)

    Clearly,

    $ Var[ni=1Li(x)]=nm1/2nVar[(Nn(x,t)E(Nn(x,t)))r(x,t)(fn(x)E(fn(x)))]=nm1/2n[Var(Nn(x,t))+r2(x,t)Var(fn(x))r(x,t)Cov(Nn(x,t),fn(x))].
    $

    Hence,

    $ \operatorname{Var}\left[\sum\limits_{i = 1}^n L_i(x)\right] = \operatorname{Var}\left(\rho(y-t)^{2}|X = x\right) f(x) \psi(x)+o(1). $

    Therefore, to complete the proof of this lemma, it is enough to show that the numerator of (A.12) converges to 0. For this, we use the $ C_r $-inequality (cf. Loève [37], page 155) to show that

    $ \sum\limits_{i = 1}^n \mathbb{E}\left[\left|L_i(x)-\mathbb{E}\left[L_i(x)\right]\right|^{2+\delta}\right] \leq C_1 \sum\limits_{i = 1}^n \mathbb{E}\left[\left|L_i(x)\right|^{2+\delta}\right]+C_2 \sum\limits_{i = 1}^n\left|\mathbb{E}\left[L_i(x)\right]\right|^{2+\delta}. $

    Recall that, because of Assumption (H4) and Lemma A.1 (ⅱ), we have

    $ ni=1E[|Li(x)|2+δ]=nδ/2(mn)34δ+32[|ρ(Yit)r(x,t)|2+δ(mn1k=0I{kmn<Xik+1mn}Bk(mn1,x))2+δ]nδ/2(mn)34δ+32mn1k=0k+1mnkmn(21+δR|ρ(Yt)|(2+δ)g(z,y)dy+21+δ|r(x,t)|2+δ)dzB2+δk(mn1,x)nδ/2(mn)34δ+32mn1k=0CmnB2+δk(mn1,x)nδ/2(mn)34δ+32×Cm32nC(m32nn)δ20.
    $

    Similarly, for the second term $ \left(\sum_{i = 1}^n\left|\mathbb{E}\left[L_i(x)\right]\right|^{2+\delta}\right) $, we get

    $ \sum\limits_{i = 1}^n\left|\mathbb{E}\left[L_i(x)\right]\right|^{2+\delta} \leq C\left(\frac{m_n^{\frac{3}{2}}}{n}\right)^{\frac{\delta}{2}} \rightarrow 0. $

    Finally, (A.9) in Lemma A.6, Lemma A.7, and Slutsky's theorem complete the proof of part 3.4 of Proposition 3.2.

    Now, if $ n m_n^{-5 / 2} \rightarrow \infty $, we have

    $ mn[(Nn(x,t)E(N(x,t)))r(x,t)(fn(x)E(fn(x)))]=(n1/2m5/4n)n1/2m1/4n[(Nn(x,t)E(Nn(x,t)))r(x,t)(fn(x)E(fn(x)))].
    $

    Since we have $ n^{-1 / 2} m_n^{5 / 4} \rightarrow 0, $ A.10 in Lemma A.6, Lemma A.7, and Slutsky's theorem complete the proof of part 3.5. Proposition 3.2 follows from (A.11) when $ x \in\{0, 1\} $.

    Lemma A.8. Under Assumptions (H1)–(H4), and for $ x \in\{0, 1\} $ such that $ f(x) > 0 $, we have:

    i) if $ m_n $ is chosen such that $ n m_n^{-3} \rightarrow c $ for some constant $ c \geq 0 $, then

    $ n1/2m1/2nfn(x)[(E(Nn(x,t))N(x,t))r(x,t)(E(fn(x))f(x))]PcΔ(x),
    $
    (A.13)

    ii) if $ m_n $ is chosen such that $ n m_n^{-3} \rightarrow \infty $, then

    $ mnfn(x)[(E(Nn(x,t))N(x,t)))r(x,t)(E(fn(x))f(x))]PΔ(x).
    $
    (A.14)

    Proof. The proof of this lemma is analogous to Lemma A.6.

    Lemma A.9. Under Assumptions (H1)–(H4), and for $ x \in\{0, 1\} $ such that $ f(x) > 0 $, we have

    $ n1/2m1/2n[(Nn(x,t)E(Nn(x,t)))r(x,t)(fn(x)E(fn(x)))]DN(0,Var(ρ(Yit))f(x)).
    $
    (A.15)

    Proof. We write

    $ n^{1 / 2} m_n^{-1 / 2}\left[\left(N_{n}(x,t)-\mathbb{E}\left(N_{n}(x,t)\right)\right)-r(x,t)\left(f_{n}(x)-\mathbb{E}\left(f_{n}(x)\right)\right)\right] = \sum\limits_{i = 1}^n\left(L_i(x)-\mathbb{E}\left(L_i(x)\right)\right), $

    where

    $ L_i(x): = \frac{m_n^{1 / 2}}{n^{1 / 2}}\left(\rho(Y_{i}-t)-r(x,t) \right) \sum\limits_{k = 0}^{m_n-1} \mathbb{I}_{\left\{\frac{k}{m_n} < X_i \leq \frac{k+1}{m_n}\right\}} B_k\left(m_n-1, x\right). $

    The proof of this lemma is based on the Lyapounov central limit theorem (FELLER, W.[36]) on $ L_i(x) $. Clearly,

    $ Var[ni=1Li(x)]=nm1nVar[(Nn(x,t)E(Nn(x,t))))r(x,t)(fn(x)E(fn(x))))]=nm1n[Var(Nn(x,t))+r2(x,t)Var(fn(x))2r(x,t)Cov(Nn(x,t),fn(x))].
    $

    Hence,

    $ \operatorname{Var}\left[\sum\limits_{i = 1}^n L_i(x)\right] = \operatorname{Var}(\rho(Y_{i}-t))f(x)+o(1). $

    Therefore, to complete the proof of this lemma, we follow the same steps as in the proof Lemma A.7, and find that

    $ \sum\limits_{i = 1}^n \mathbb{E}\left[\left|L_i(x)-\mathbb{E}\left[L_i(x)\right]\right|^{2+\delta}\right] \leq \frac{C}{m_n^{\frac{1}{2}}} \times\left(\frac{m_n}{n}\right)^{\frac{\delta}{2}} \rightarrow 0 . $

    Finally, (A.13) in Lemma A.8, Lemma A.9, and Slutsky's theorem complete the proof of part 3.6 of Proposition 3.2. Now, if $ n m_n^{-3} \rightarrow \infty $, we have

    $ mn[(Nn(x,t)E(Nn(x,t)))r(x,t)(fn(x)E(fn(x)))]=(n1/2m3/2n)n1/2m1/2n[(Nn(x,t)E(Nn(x,t)))r(x,t)(fn(x)E(fn(x)))].
    $

    Since we have $ n^{-1 / 2} m_n^{3 / 2} \rightarrow 0, $ A.14 in Lemma A.8, Lemma A.9, and Slutsky's theorem completes the proof of part 3.7 of Proposition 3.2 follows from (A.15).

    First, we have

    $ 10Bias(ˆrn(x,t))2dx=10(E[ˆrn(x,t)]r(x,t))2dx=10Δ2(x)m2n+o(1m2n)dx=δ1m2n+o(1m2n).
    $

    Moreover, we have

    $ \operatorname{Var}\left(\widehat{r}_n(x,t)\right) = \left\{\frac{1}{f^2(x)} \operatorname{Var}\left(N_{n}(x,t)\right)+\frac{r^2(x,t)}{f^2(x)} \operatorname{Var}\left(f_n(x)\right)-2 \frac{r(x,t)}{f^2(x)} \operatorname{Cov}\left(N_{n}(x,t), f_n(x)\right)\right\}[1+o(1)]. $

    Then,

    $ 10Var(ˆrn(x,t))dx={10Var(Nn(x,t))f2(x)dx+10r2(x,t)Var(fn(x))f2(x)dx210r(x,t)Cov(Nn(x,t),fn(x))f2(x)dx}[1+o(1)].
    $
    (A.16)

    First, we have

    $ \operatorname{Var}\left[f_{n}(x)\right] = \frac{1}{n}\left[A_{m}(x)-f_{m}^{2}(x)\right], $

    where $ f_{m}^{2}(x) = \mathbb{E}^{2}\left[f_{n}(x)\right] = f^{2}(x)+O\left(m_{n}^{-1}\right) $, and

    $ Am(x)=m2nmn1k=0[F(k+1m)F(km)]B2k(mn1,x)=mn[f(x)Sm1(x)+O(Hm1(x))+O(m1)],
    $

    for $ x \in[0, 1] $ and $ m_{n} \geq 2 $, where

    $ H_m(x) = \sum\limits_{k = 0}^m\left|\frac{k}{m}-x\right| B_k^2\left(m_n, x\right) = O_x\left(m_{n}^{-3 / 4}\right) . $

    Note that this error term is not uniform. For this, we use the Cauchy-Schwarz inequality to write

    $ Hmn(x)[mnk=0(kmnx)2Bk(mn,x)]1/2[mnk=0B3k(mn,x)]1/2[Smn(x)4mn]1/2,
    $
    (A.17)

    for all $ m_{n} \geq 1 $ and $ x \in[0, 1] $, since $ 0 \leq B_k\left(m_n, x\right) \leq 1 $ and

    $ \sum\limits_{k = 0}^{m_{n}}\left(\frac{k}{m_{n}}-x\right)^2 B_k\left(m_n, x\right) = \frac{x(1-x)}{m_{n}} \leq \frac{1}{4 m_{n}} . $

    Then, starting from Eq (A.17) and applying Jensen's inequality and Lemma A.1 (ⅳ), we have

    $ 10g(x)Hmn(x)dx10g(x)[Smn(x)4mn]1/2dx[10g(x)dx]1/2[14m3/2n10g(x)ψ(x)dx+o(m3/2n)]1/2=O(m3/4n).
    $

    Then, we infer that

    $ 10r2(x,t)Var[fn(x)]{f(x)}2dx=1n10r2(x,t)Amn(x)f2mn(x){f(x)}2dx=1n[10r2(x,t)Amn(x){f(x)}2dx10r2(x,t)]+O(1mn)=mnn[10r2(x,t){f(x)}2(Smn1(x)+O(Hmn1(x))+O(m1n))dx]1n10r2(x,t)+O(1mn)=mnn[10r2(x,t)f(x)Smn1(x)dx+O(m3/4n)]1n10r2(x,t)+O(1mn),
    $

    and, using Lemma A.1 (ⅳ), we have

    $ 10r2(x,t)Var[fn(x)]{f(x)}2dx=m1/2nn10r2(x,t)f(x)ψ(x)dx1n10r2(x,t)+o(m1/2nn)+O(1mn).
    $
    (A.18)

    Second, we have

    $ Cov[fn(x),Nn(x,t)]=1n{m2nmn1k=0(k+1mkmnr(z)f(x)dz)B2k(mn1,x)E[fn(x)]E[Nn(x,t)]}=m2nnmn1k=0(k+1mnkmn[r(x,t)f(x)+O(zx)]dz)B2k(mn1,x)1nf(x)N(x,t)+O(1mn)=mnn[r(x,t)f(x)Smn1(x)+O(Hmn1(x))+O(m1n)]1nf(x)N(x,t)+O(1mn).
    $

    Then, using the same argument for $ H_{mn-1}(x) $ as previously, we obtain

    $ 10r(x,t)Cov[fn(x),Nn(x,t)]{f(x)}2dx=mnn[10r2(x,t)f(x)Smn1(x)dx+O(m3/4n)]1n10r2(x,t)+O(1mn)=m1/2nn10r2(x,t)f(x)ψ(x)dx1n10r2(x,t)+o(m1/2nn)+O(1mn).
    $
    (A.19)

    Third, we have

    $ Var[Nn(x,t)]=m2nnmn1k=0(k+1mnkmnE[ρ(Yt)2X=z]f(z)dz)B2k(mn1,x)1nE2[Nn(x,t)]=m2nnmn1k=0(k+1mnkmn[E[ρ(Yt)2X=x]f(x)+O(zx)]dz)B2k(mn1,x)1nN2(x,t)+O(1mn)=mnn[E[ρ(Yt)2X=x]f(x)Smn1(x)+O(Hmn1(x))+O(m1n.)]1nN2(x,t)+O(1mn).
    $

    Then,

    $ 10Var[Nn(x,t)]{f(x)}2dx=mnn[10E[ρ(Yt)2X=x]f(x)Smn1(x)dx+O(m3/4n)]1n10r2(x,t)+O(1mn)=m1/2nn10E[ρ(Yt)2X=x]f(x)ψ(x)dx1n10r2(x,t)+o(m1/2nn)+O(1mn).
    $
    (A.20)

    Finally, substituting (A.18), (A.19), and (A.20) into (A.16), we obtain

    $ 10Var[ˆrn(x,t)]dx=(10E[ρ(Yt)2X=x]f(x)ψ(x)dx10E2[ρ(Yt)X=x]f(x)ψ(x)dx)m1/2nn+o(m1/2nn)=10E[ρ(Yt)2X=x]E2[ρ(Yt)X=x]f(x)ψ(x)dxm1/2nn+o(m1/2nn)=10Var[ρ(Yt)X=x]f(x)ψ(x)dxm1/2nn+o(m1/2nn).
    $

    Then, we obtain

    $ MISE(ˆrn)=10{Var(ˆrn(x,t))+Bias2(ˆrn(x,t))}=Λ1m2n+Λ2m1/2nn+o(m1/2nn)+o(m2n).
    $

    Using a Taylor expansion of order one around $ \theta $, we get

    $ \widehat{r}(x, \widehat{\theta_{x}}) = \widehat{r}(x, \theta_{x})+(\widehat{\theta_{x}}-\theta_{x}) \frac{\partial \widehat{r}}{\partial t}\left(x, \xi_{n}\right), $

    with $ \xi_{n} \in(\widehat{\theta_{x}}, \theta_{x}) $. Because of the definition of $ \widehat{\theta} $, we have

    $ \widehat{\theta_{x}}-\theta_{x} = \frac{-\widehat{r}(x, \theta_{x})}{\frac{\partial \hat{r}}{\partial t}\left(x, \xi_{n}\right)}. $

    We will prove that the numerator is asymptotically normal, whereas the denominator converges in probability to $ \Gamma(x, \theta_{x}) $; for that, we will use the following decompositions:

    i) When $ x\in (0, 1) $ and $ m_{n} $ is chosen such that $ n m^{-5/2}\rightarrow c $, then

    $ n^{1 / 2} m_n^{-1 / 4} (\widehat \theta_{x}-\theta_{x}) = \dfrac{-n^{1 / 2} m_n^{-1 / 4}[\widehat{r}(x,\theta_{x})-r(x,\theta_{x})]}{\frac{\partial \widehat r}{\partial t}\left(x, \xi_{n}\right)}. $

    ii) When $ x\in \{0, 1\} $ and $ m_{n} $ is chosen such that $ n m^{-3}\rightarrow c $, then

    $ n^{1 / 2} m_n^{-1 / 2} (\widehat \theta_{x}-\theta_{x}) = \dfrac{-n^{1 / 2} m_n^{-1 / 2}[\widehat{r}(x,\theta_{_{x}})-r(x,\theta_{x})]}{\frac{\partial \widehat r}{\partial t}\left(x, \xi_{n}\right)}. $

    So, we state asymptotic normality by Slutsky's Theorem, and by Proposition 3.2 with $ t = \theta $. We show that the numerator suitably normalized is asymptotically normally distributed. Then, it suffices to show that the denominator converges in probability to $ \Gamma(x, \theta_{x}) $ (see Lemma A.10).

    Lemma A.10. Under Assumptions (H1)–(H3), and for $ x \in[0, 1] $ where $ f(x) > 0 $, we have

    $ \frac{\partial \widehat{r}}{\partial t}\left(x, \xi_{n}\right) \xrightarrow{\mathbb{P}} \Gamma(x, \theta_{x} ). $

    Proof. We explore the following decomposition:

    $ |ˆrt(x,ξn)Γ(x,θx)||ˆrt(x,ξn)ˆrt(x,θx)|+|ˆrt(x,θx)Γ(x,θx)|J1(x)+J2(x).
    $
    (A.21)

    For $ J_{1}(x) $, we write

    $ J1(x)supy[a,b]|ρ(yξn)tρ(yθx)t|mnfn(x)nni=1m1k=0I{km<Xik+1m}Bk(m1,x).
    $

    Because $ \frac{\partial \rho(y-t)}{\partial t} $ is continuous at $ \theta $ uniformly, the use of Theorem 3.1 and the convergence in probability of $ f_{n}(x) $ to $ f(x) $ show that the first term of (A.21) converges in probability to 0. However, the limit of the second term is obtained by evaluating, separately, the bias and the variance terms of $ \frac{\partial \widehat{r}}{\partial t}\left(x, \theta_{x} \right) $. Clearly, a similar argument to those invoked for proving (3.1) can be used to obtain that

    $ \frac{\partial \widehat{r}}{\partial t}(x, \theta_{x} )\rightarrow \Gamma(x, \theta_{x} ) \quad \text{in probability}. $

    [1] Xu L, Chi I. (2011) Life satisfaction among rural Chinese grandparents: the roles of intergenerational family relationship and support exchange with grandchildren. Int J Social Welfare 20(s1): S148-S159.
    [2] Chow N. (2004) Asian value and aged care. Geriatr Gerontol Int 4(S1): S21-S25.
    [3] Kim U, Tirandis H, Kagitcibasi G, et al. (Eds. ) (1994) Individualism and collectivism: theory, method, and applications Thousand Oaks: Sage.
    [4] Chen F, Liu G. (2012) The health implications of grandparents caring for grandchildren in China. J Gerontol 67(1): 99-112.
    [5] Xie X, Xia Y. (2011) Grandparenting in Chinese Immigrant Families. Marriage Family Rev 47(6):383-396.
    [6] Bennett P, Meredith WH. (1995) The modern role of the grandmother in China: a departure from the confucian ideal. Int J Sociol Family 25(1): 1-12.
    [7] Bowers B, Myers B. (1999) Grandmothers providing care for grandchildren: consequences of various levels of caregiving. Family Relat 48(3): 303-311.
    [8] Pruchno RA, McKenney D. (2002) Psychological well-being of black and white grandmothers raising grandchildren: examination of a two-factor model. J Gerontol 57B(5): 444-452.
    [9] Waldrop DP, Weber JA. (2001) From grandparent to caregiver: the stress and satisfaction of raising grandchildren. Families Society: J Contemp Human Services 82(5): 461-472.
    [10] Musil CM, Ahmad M. (2002) Health of grandmothers a comparison by caregiver status. J Aging Health 14(1): 96-121.
    [11] Musil CM. (2000) Health of grandmothers as caregivers: A ten month follow-up. J Women Aging 12(1-2): 129-145.
    [12] Dong X, Simon M, Evans D. (2012) Decline in physical function and risk of elder abuse reported to social services in a community-dwelling population of older adults. J Am Geriatr Society 60(10): 1922-1928.
    [13] Dong X, Chang ES, Wong E, et al. (2010) Assessing the health needs of Chinese older adults: findings from a community-based participatory research study in Chicago’s Chinatown. J Aging Res 2010: 1-12.
    [14] Kamo Y. (1998) Asian grandparents. In: Handbook on grandparenthood. New York: Greenwood Press, 97-112.
    [15] Lee E. (1997) Chinese American families. In: Working with Asian Americans: A guide for clinicians: 46-78.
    [16] Kramer EJ, Kwong K, Lee E, et al. (2002) Cultural factors influencing the mental health of Asian Americans. Western J Med 176(4): 227.
    [17] Tam VCW, Detzner D. (1998) Grandparents as a family resource in Chinese-American families: Perceptions of the middle generation. In: MCCubbin HI, Thompson EA, Thompson AI, et al (Eds. ), Resiliency in Native-American and immigrant families. Thousand Oaks: Sage, 243-263.
    [18] Phinney J, Ong A, Madden T. (2000) Cultural values and intergenerational value discrepancies in immigrant and non-immigrant Families. Child Dev 71(2): 528-539.
    [19] Taylor P, Passel J, Fry R, et al. (2010) The return of the multi-generational family household. Pew Res Center 18: 1-25.
    [20] Silverstein M, Chen X. (1999) The impact of acculturation in Mexican American families on the quality of adult grandchild-grandparent relationships. J Marriage Family 61(1): 188-198.
    [21] Dong X, Chang ES, Wong E, et al. (2012) A qualitative study of filial piety among community dwelling, Chinese, older adults: Changing meaning and impact on health and well-being. J Intergenerat Relat 10(2): 131-146.
    [22] Chen R, Simon MA, Chang ES, et al. (2014) The perception of social support among U. S. Chinese older adults: findings from the PINE study. J Aging Health 26(7): 1137-1154.
    [23] Lou V. (2010) Life satisfaction of older adults in Hong Kong: The role of social support from grandchildren. Social Indic Res 95: 377-391. doi: 10.1007/s11205-009-9526-6
    [24] Lo M, Liu YH. (2009) Quality of life among older grandparent caregivers: a pilot study. J Adv Nurs 65(7): 1475-1484.
    [25] Treas J, Mazumdar S. (2002) Older people in America’s immigrant families: Dilemmas of dependence, integration, and isolation. J Aging Studies 16: 243-258. doi: 10.1016/S0890-4065(02)00048-8
    [26] Guo B, Pickard J, Huang J. (2008) A cultural perspective on health outcomes of caregiving grandparents: Evidence from China. J Intergen Relat 5(4): 25-40.
    [27] Goh EC. (2009) Grandparents as childcare providers: An in-depth analysis of the case of Xiamen, China. J Aging Studies 23(1): 60-68.
    [28] Chen F, Liu G, Mair CA. (2011) Intergenerational ties in context: Grandparents caring for grandchildren in China. Social Forces 90(2): 571-594.
    [29] Ko PC, Hank K. (2014) Grandparents caring for grandchildren in China and Korea: Findings from CHARLS and KLoSA. J Gerontol Series B: Psychol Sci Social Sci 69(4): 646-651.
    [30] Sun J. (2013) Chinese older adults taking care of grandchildren: Practices and policies for productive aging. Ageing Int 38(1): 58-70.
    [31] Wei-Qun LV, Chi I. (2008) Measuring grandparenthood stress and reward: Developing a scale based on perceptions by grandparents with adolescent grandchildren in Hong Kong. Geriatr Gerontol Int 8: 291-299. doi: 10.1111/j.1447-0594.2008.00484.x
    [32] Strom RD, Strom SK, Wang CM, et al. (1999) Grandparents in the United States and the Republic of China: A comparison of generations and cultures. Int J Aging Human Develop 49(4):279-317.
    [33] Chen J, Murayama S, Kamibeppu K. (2010) Factors related to well-being among the elderly in urban China focusing on multiple roles. Biosci Trends 4: 61-71.
    [34] Grinstead LN, Leder S, Jensen S, et al. (2003) Review of research on the health of caregiving grandparents. J Adv Nursing 44(3): 318-326.
    [35] Hayslip B, Kaminski PL. (2005) Grandparents raising their grandchildren: A review of the literature and suggestions for practice. Gerontologist 45(2): 262-269.
    [36] Kataoka-Yahiro M, Ceria C, Caulfield R. (2004) Grandparent caregiving role in ethnically diverse families. J Pediatr Nursing 19(5): 315-328.
    [37] Nagata D, Cheng WJY, Tsai-Chae AH. (2010) Chinese American grandmothering: a qualitative exploration. Asian Am J Psychol 1(2): 151-161.
    [38] Pruchno R. (1999) Raising grandchildren: The experiences of black and white grandmothers. Gerontologist 39(2): 209-221.
    [39] Simon MA, Magee M, Shah A, et al. (2008) Building a Chinese community health survey in Chicago: the value of involving the community to more accurately portray health. Int J Health Ageing Manag 2(1): 42-57.
    [40] Dong X, Wong E, Simon M. (2014) Study design and implementation of the PINE study. J Aging Health. Published online 25 March 2014.
    [41] Dong XQ, Chang ES, Wong E, et al. (2011) Working with culture: lessons learned from a community-engaged project in a Chinese aging population. Aging Health 7(4): 529-537.
    [42] Dong XQ, Chang ES, Simon M, et al. (2011) Sustaining community-university partnerships: lessons learned from a participatory research project with elderly Chinese. Gateways: Int J Commun Res Engag 4: 31-47. doi: 10.5130/ijcre.v4i0.1767
    [43] Dong X, Li Y, Chen R, et al. (2013) Evaluation of community health education workshops among Chinese older adults in Chicago: a community-based participatory research approach. J Educ Train Studies 1(1): 170-181.
    [44] Chang ES, Simon M, Dong X. (2012) Integrating cultural humility into health care professional education and training. Adv Health Sci Educ 17(2): 269-278.
    [45] Simon M, Chang E, Rajan K, et al. (2014) Demographic characteristics of U. S. Chinese older adults in the greater Chicago area: Assessing the representativeness of the PINE study. J Aging Health In press.
    [46] Musil CM. (1998) Health, stress, coping, and social support in grandmother caregivers. Health Care Women Int 19(5): 441-455.
    [47] Luo Y, LaPierre TA, Hughes ME, et al. (2012) Grandparents providing care to grandchildren: a population-based study of continuity and change. J Family Issues 33(9): 1143-1167.
    [48] Lumpkin JR. (2008) Grandparents in a parental or near-parental role: Sources of stress and coping mechanisms. J Family Issues 29(3): 357-372.
    [49] Ku LJE, Stearns SC, Van Houtven CH, et al. (2013) Impact of caring for grandchildren on the health of grandparents in Taiwan. J Gerontol Series B: Psychol Sci Social Sci 68(6): 1009-1021.
    [50] Minkler M, Fuller-Thomson E, Miller D, et al. (1997) Depression in grandparents raising grandchildren. Arch Family Med 6: 445-452. doi: 10.1001/archfami.6.5.445
    [51] Hughes ME, Wait LJ, LaPierre TA, et al. (2007) All in the family: the impact of caring for grandchildren on grandparents’ health. J Gerontol: Social Sci 62B(2): S108-119.
    [52] Burton LM. (1992) Black grandparents rearing children of drug-addicted parents: Stressors, outcomes, and social service needs. Gerontologist 32: 744-751. doi: 10.1093/geront/32.6.744
    [53] Kelley SJ. (1993) Caregiver stress in grandparents raising grandchildren. IMAGE: J Nursing Scholar 25: 331-337. doi: 10.1111/j.1547-5069.1993.tb00268.x
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