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Uniform convergence of estimator for nonparametric regression with dependent data
Journal of Inequalities and Applications volume 2016, Article number: 142 (2016)
Abstract
In this paper, the authors investigate the internal estimator of nonparametric regression with dependent data such as α-mixing. Under suitable conditions such as the arithmetically α-mixing and \(E|Y_{1}|^{s}<\infty\) (\(s>2\)), the convergence rate \(|\widehat{m}_{n}(x)-m(x)|=O_{P}(a_{n})+O(h^{2})\) and uniform convergence rate \(\sup_{x\in S_{f}^{\prime}}|\widehat{m}_{n}(x)-m(x)|=O_{p}(a_{n})+O(h^{2})\) are presented, if \(a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\rightarrow0\). We generalize some results in Shen and Xie (Stat. Probab. Lett. 83:1915-1925, 2013).
1 Introduction
Kernel-type estimators of the regression function are widely various situations because of their flexibility and efficiency, in the dependent cases as well as in the independent data case. This paper is concerned with the nonparametric regression model
where \((X_{i},Y_{i})\in R^{d}\times R\), \(1\leq i\leq n\), \(U_{i}\) is random variable such that \(E(U_{i}|X_{i})=0\), \(1\leq i\leq n\). Then one has
The most popular nonparametric estimators of the unknown function \(m(x)\) is the Nadaraya-Watson estimator \(\widehat{m}_{NW}(x)\) given below and the local polynomials fitting. Let \(K(x)\) be a kernel function. Define \(K_{h}(x)=h^{-d}K(x/h)\), where \(h=h_{n}\) is a sequence of positive bandwidths tending to zero as \(n\rightarrow\infty\). Kernel-type estimators of the regression function are widely various situations because of their flexibility and efficiency, in the dependent data case as well as the independent data case. For the independent data, Nadaraya [2] and Watson [3] gave the most popular nonparametric estimators of the unknown function \(m(x)\), named the Nadaraya-Watson estimator \(\widehat{m}_{NW}(x)\), i.e.
Jones et al. [4] considered various versions of kernel-type regression estimators such as the Nadaraya-Watson estimator (1.2) and the local linear estimator. They also investigated the following internal estimator:
for known density \(f(\cdot)\). The term ‘internal’ stands for the fact that the factor \(\frac{1}{f(X_{i})}\) is internal to the summation, while the estimator \(\widehat{m}_{NW}(x)\) has the factor \(\frac{1}{\widehat{f}(x)}=\frac{1}{n^{-1}\sum_{i=1}^{n}K_{h}(x-X_{i})}\) externally to the summation.
The internal estimator was first proposed by Mack and Müller [5]. Jones et al. [4] studied various kernel-type regression estimators, including introduced the internal estimator (1.3). Linton and Nielsen [6] introduced ‘integration method’, based on direct integration of initial pilot estimator (1.3). Linton and Jacho-Chávez [7] studied the two internal nonparametric estimators with the estimator similar to estimator (1.3) but in place of an unknown density \(f(\cdot)\), a classical kernel estimator \(\widehat{f}(x)=n^{-1}\sum_{i=1}^{n} K_{h}(x-X_{i})\) is used. Much work has been done for the kernel density estimation. For example, Masry [8] gave the recursive probability density estimation for a mixing dependent sample, Roussas et al. [9] and Tran et al. [10] investigated the fixed design regression for dependent data, Liebscher [11] studied the strong convergence of sums of α-mixing random variables and gave its application to density estimation, Hansen [12] obtained the uniform convergence rates for kernel estimation with dependent data, and so on. For more work as regards kernel estimation, we can also refer to [13–30] and the references therein.
Let \((\Omega, \mathcal{F}, P)\) be a fixed probability space. Denote \(N=\{1,2,\ldots,n,\ldots\}\). Let \(\mathcal {F}_{m}^{n}=\sigma(X_{i}, m\leq i\leq n, i\in N)\) be the σ-field generated by random variables \(X_{m},X_{m+1},\ldots,X_{n}\), \(1\leq m\leq n\). For \(n\geq1\), we define
Definition 1.1
If \(\alpha(n)\downarrow0\) as \(n\rightarrow \infty\), then \(\{X_{n},n\geq1\}\) is called a strong mixing or α-mixing sequence.
Recently, Shen and Xie [1] obtained the strong consistency of the internal estimator (1.3) under α-mixing data. In their paper, the process is assumed to be geometrically α-mixing sequence, i.e. the mixing coefficients \(\alpha(n)\) satisfy \(\alpha(n)\leq\beta_{0}e^{-\beta_{1}n}\), where \(\beta_{0}>0\) and \(\beta_{1}>0\). They also supposed that the sequence \(\{Y_{n},n\geq 1\}\) is bounded, as well as the density \(f(x)\) of \(X_{1}\). Inspired by Hansen [12], Shen and Xie [1] and other papers above, we also investigate the convergence of the internal estimator (1.3) under α-mixing data. The process is supposed to be an arithmetically α-mixing sequence, i.e. the mixing coefficients \(\alpha(n)\) satisfy \(\alpha(n)\leq Cn^{-\beta}\), \(C>0\), and \(\beta>0\). Without the bounded conditions of \(\{Y_{n},n\geq 1\}\) and the density \(f(x)\) of \(X_{1}\), we establish the convergence rate and uniform convergence rate for the internal estimator (1.3). For the details, please see our results in Section 2. The conclusion and the lemmas and proofs of the main results are presented in Section 3 and Section 4, respectively.
Regarding notation, for \(x=(x_{1},\ldots,x_{d})\in R^{d}\), set \(\|x\|=\max(|x_{1}|,\ldots,|x_{d}|)\). Throughout the paper, \(C,C_{1},C_{2},C_{3},\ldots\) , denote some positive constants not depending on n, which may be different in various places. \(\lfloor x\rfloor\) denotes the largest integer not exceeding x. → means to take the limit as \(n\rightarrow\infty\), \(\xrightarrow{P}\) means to convergence in probability. \(X\stackrel{d}{=}Y\) means that the random variables X and Y have the same distribution. A sequence \(\{X_{n},n\geq1\}\) is said to be of second-order stationarity if \((X_{1},X_{1+k})\stackrel{d}{=} (X_{i},X_{i+k})\), \(i\geq1\), \(k\geq1\).
2 Results and discussion
2.1 Some assumptions
Assumption 2.1
We assume the data observed \(\{(X_{n},Y_{n}),n\geq1\}\) valued in \(R^{d}\times R\) comes from a second-order stationary stochastic sequence. The sequence \(\{(X_{n},Y_{n}),n\geq1\}\) is also assumed to be arithmetically α-mixing with mixing coefficients \(\alpha(n)\) such that
where \(A<\infty\) and for some \(s>2\)
and
The known density \(f(\cdot)\) of \(X_{1}\) is upon its compact support \(S_{f}\) and it is also assumed that \(\inf_{x\in S_{f}}f(x)>0\). Let \(B_{0}\) be a positive constant such as
Also, there is a \(j^{*}<\infty\) such that for all \(j\geq j^{*}\)
where \(B_{1}\) is a positive constant and \(f_{j}(x_{1},x_{j+1})\) denotes the joint density of \((X_{1},X_{j+1})\).
Assumption 2.2
There exist two positive constants \(\bar{K}>0\) and \(\mu>0\) such that
Assumption 2.3
Denote by \(S_{f}^{0}\) the interior of \(S_{f}\). For \(x\in S_{f}^{0}\), the function \(m(x)\) is twice differentiable and there exists a positive constant b such that
The kernel density function \(K(\cdot)\) is symmetrical and satisfies
Assumption 2.4
The kernel function satisfies the Lipschitz condition, i.e.
Remark 2.1
Similar to Assumption 2 of Hansen [12], Assumption 2.1 specifies that the serial dependence in the data is of strong mixing type, and equations (2.1)-(2.3) specify a required decay rate. Condition (2.4) controls the tail behavior of the conditional expectation \(E(|Y_{1}|^{s}|X_{1}=x)\), condition (2.5) places a similar bound on the joint density and conditional expectation. Assumptions 2.2-2.4 are the conditions of kernel function \(K(u)\), i.e., Assumption 2.2 is a general condition, Assumption 2.3 is used to estimate the convergence rate of \(|E\widehat{m}_{n}(x)-m(x)|\), and Assumption 2.4 is used to investigate the uniform convergence rate of the internal estimator \(\widehat{m}_{n}(x)\).
2.2 Main results
First, we investigate the variance bound of estimator \(\widehat{m}_{n}(x)\). For \(1\leq r\leq s\) and \(s>2\), denote \(\bar{\mu}(r,s):=\frac{(B_{0})^{r/s}\bar{K}^{r-1}\mu}{(\inf_{x\in S_{f}} f(x))^{r-1+r/s}}\), where \(B_{0}\), \(\inf_{x\in S_{f}} f(x)\), K̄, and μ are defined in Assumptions 2.1 and 2.2.
Theorem 2.1
Let Assumption 2.1 and Assumption 2.2 be fulfilled. Then there exists a \(\Theta<\infty\) such that for n sufficiently large and \(x\in S_{f}\)
where \(\Theta:=\bar{\mu}(2,s)+2j^{*}\bar{\mu}(2,s)+2 (\bar{\mu}^{2}(1,s)+\frac {B_{1}\mu^{2}}{(\inf_{x\in S_{f}}f)^{2}} )+\frac{16A^{1-2/s}\bar{\mu}^{\frac{2}{s}}(s,s)}{(s-2)/s}\).
As an application to Theorem 2.1, we obtain the weak consistency of estimator \(\widehat{m}_{n}(x)\).
Corollary 2.1
Let Assumption 2.1 and Assumption 2.2 be fulfilled and \(K(\cdot)\) be a density function. For \(x\in S_{f}\), \(m(x)\) is supposed to be continuous at x. If \(nh^{d}\rightarrow \infty\) as \(n\rightarrow\infty\), then
Next, the convergence rate of estimator \(\widehat{m}_{n}(x)\) is presented.
Theorem 2.2
For \(0<\theta<1\) and \(s>2\), let Assumptions 2.1-2.3 hold, where the mixing exponent β satisfies
Denote \(a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\) and take \(h=n^{-\theta/d}\). Then for \(x\in S_{f}^{0}\), one has
Third, we now investigate the uniform convergence rate of estimator \(\widehat{m}_{n}(x)\) and its convergence over a compact set. Let \(S_{f}^{\prime}\) be any compact set contained in \(S_{f}^{0}\).
Theorem 2.3
For \(0<\theta<1\) and \(s>2\), let Assumptions 2.1-2.3 be fulfilled, where the mixing exponent β satisfies
Suppose that Assumption 2.4 is also fulfilled. Denote \(a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\) and take \(h=n^{-\theta/d}\). Then
2.3 Discussion
The parametric θ in Theorem 2.2 and Theorem 2.3 plays the role of a bridge between the process (i.e. mixing exponent) and choice of positive bandwidth h. For example, if \(d=2\), \(\theta=\frac{1}{3}\), and \(\beta>\max\{\frac{s+1}{s-2},\frac{2s-2}{s-2}\}\), then we take \(h=n^{-1/6}\) in Theorem 2.2 and obtain the convergence rate \(|\widehat{m}_{n}(x)-m(x)|= O_{P}((\ln n)^{1/2}n^{-1/3})\). Similarly, if \(d=2\), \(\theta=\frac{1}{3}\), and \(\beta>\frac{2s-2}{s-2}\), then we choose \(h=n^{-1/6}\) in Theorem 2.3 and establish the uniform convergence rate \(\sup_{x\in S_{f}^{\prime}}|\widehat{m}_{n}(x)-m(x)|=O_{p}((\ln n)^{1/2}n^{-1/3})\).
3 Conclusion
On the one hand, similar to Theorem 2.1, Hansen [12] investigated the kernel average estimator
and obtained the variance bound \(\operatorname{Var}(\widehat{\Psi}(x))\leq \frac{\Theta}{nh^{d}}\), where Θ is a positive constant. For the details, see Theorem 1 of Hansen [12]. Under some other conditions, Hansen [12] also gave the uniform convergence rates such as \(\sup_{\|x\|\leq c_{n}}|\widehat{\Psi}(x)-E\widehat{\Psi}(x)|=O_{P}(a_{n})\), where \(a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\rightarrow0\) and \(\{c_{n}\}\) is a sequence of positive constant (see Theorems 2-5 of Hansen [12]). On the other hand, under the conditions such as the geometrically α-mixing and \(\{Y_{n}, n\geq1\}\) is bounded as well as the density function \(f(x)\) of \(X_{1}\), Shen and Xie [1] obtained the complete convergence such as \(|\widehat{m}_{n}(x)-m(x)|\xrightarrow{a.c.}0\), if \(\frac{\ln^{2}n}{nh^{d}}\rightarrow0\) (see Theorem 3.1 of Shen and Xie [1]), the uniform complete convergence such as \(\sup_{x\in S_{f}^{\prime}}|\widehat{m}_{n}(x)-m(x)|\xrightarrow{a.c.}0\), if \(\frac{\ln^{2}n}{nh^{d}}\rightarrow0\) (see Theorem 4.1 of Shen and Xie [1]). In this paper, we do not need the bounded conditions of \(\{Y_{n},n\geq1\}\) and \(f(x)\) of \(X_{1}\), and we also investigate the convergence of the internal estimator \(\hat{m}_{n}(x)\). Under some weak conditions such as the arithmetically α-mixing and \(E|Y_{1}|^{s}<\infty\), \(s>2\), we establish the convergence rate in Theorem 2.2 such as \(|\widehat{m}_{n}(x)-m(x)|= O_{P}(a_{n})+O(h^{2})\) if \(a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\rightarrow0\), and uniform convergence rate in Theorem 2.3 such as \(\sup_{x\in S_{f}^{\prime}}|\widehat{m}_{n}(x)-m(x)|=O_{p}(a_{n})+O(h^{2})\) if \(a_{n}=\sqrt{\frac{\ln n}{nh^{d}}}\rightarrow0\). In Theorem 2.2 and Theorem 2.3, we have \(|\widehat{m}_{n}(x)-E\widehat{m}_{n}(x)|=O_{P}(a_{n})\) and \(\sup_{x\in S_{f}^{\prime}}|\widehat{m}_{n}(x)-E\widehat{m}_{n}(x)|=O_{P}(a_{n})\), where the convergence rates are the same as that obtained by Hansen [12]. So, we relatively generalize the results in Shen and Xie [1].
4 Some lemmas and the proofs of the main results
Lemma 4.1
(Hall and Heyde, [31], Corollary A.2, i.e. Davydov’s lemma)
Suppose that X and Y are random variables which are \(\mathscr{G}\)-measurable and \(\mathscr{H}\)-measurable, respectively, and \(E|X|^{p}<\infty\), \(E|Y|^{q}<\infty\), where \(p,q>1\), \(p^{-1}+q^{-1}<1\). Then
Lemma 4.2
(Liebscher [32], Proposition 5.1)
Let \(\{X_{n},n\geq1\}\) be a stationary α-mixing sequence with mixing coefficient \(\alpha(k)\). Assume that \(EX_{i}=0\) and \(|X_{i}|\leq S<\infty\), a.s., \(i=1,2,\ldots,n\). Then, for \(n,m\in N\), \(0< m\leq n/2\), and all \(\varepsilon>0\),
where \(D_{m}=\max_{1\leq j\leq2m}\operatorname{Var}(\sum_{i=1}^{j} X_{i})\).
Lemma 4.3
(Shen and Xie [1], Lemma 3.2)
Under Assumption 2.3, for \(x\in S_{f}^{0}\), one has
Proof of Theorem 2.1
For \(x\in S_{f}\), let \(Z_{i}:=\frac{Y_{i}K_{h}(x-X_{i})}{f(X_{i})}\), \(1\leq i\leq n\). Consider now
For any \(1\leq r\leq s\) and \(s>2\), it follows from (2.4) and (2.6) that
For \(j\geq j^{*}\), by (2.5), one has
Define the covariances \(\gamma_{j}=\operatorname{Cov}(Z_{1},Z_{j+1})\), \(j>0\). Assume that n is sufficiently large so that \(h^{-d}\geq j^{*}\). We now bound the \(\gamma_{j}\) separately for \(j\leq j^{*}\), \(j^{*}< j\leq h^{-d}\), and \(h^{-d}< j<\infty\). First, for \(1\leq j\leq j^{*}\), by the Cauchy-Schwarz inequality and (4.1) with \(r=2\),
Second, for \(j^{*}< j\leq h^{-d}\), in view of (4.1) (\(r=1\)) and (4.2), we establish that
Third, for \(j>h^{-d}\), we apply Lemma 4.1, (2.1) and (4.1) with \(r=s\) (\(s>2\)) and we thus obtain
Consequently, in view of the property of second-order stationarity and (4.3)-(4.5), for n sufficiently large, we establish
where the final inequality uses the fact that \(\sum_{j=k+1}^{\infty}j^{-\delta}\leq\int _{k}^{\infty}x^{-\delta}\,dx=\frac{k^{1-\delta}}{\delta-1}\) for \(\delta>1\) and \(k\geq1\).
Thus, (2.7) is completely proved. □
Proof of Corollary 2.1
It is easy to see that
which can be treated as ‘variance’ part and ‘bias’ part, respectively.
On the one hand, by the proof of Theorem 3.1 of Shen and Xie [1], one has \(|E\widehat{m}_{n}(x)-m(x)|\rightarrow0\). On the other hand, we apply Theorem 2.1 and obtain that \(|\widehat{m}_{n}(x)-E\widehat{m}_{n}(x)|\xrightarrow{ P } 0\). So, (2.8) is proved finally. □
Proof of Theorem 2.2
Let \(\tau_{n}=a_{n}^{-1/(s-1)}\) and define
Obviously, we have
Combining Markov’s inequality with (4.7), one has
Denote
It can be seen that
Similar to the proof of (4.6), it can be argued that
which implies
Meanwhile, one has \(|\tilde{Z}_{i}-E\tilde{Z}_{i}|\leq \frac{C_{1}\tau_{n}}{h^{d}}\), \(1\leq i\leq n\). Setting \(m=a_{n}^{-1}\tau_{n}^{-1}\) and using (2.9), \(h=n^{-\theta/d}\), and Lemma 4.2 with \(\varepsilon=a_{n}n\), we obtain for n sufficiently large
in view of \(s>2\), \(0<\theta<1\), \(\beta>\max\{\frac{\theta s+s-2\theta}{(1-\theta)(s-2)},\frac{2s-2}{s-2}\}\), and \(\frac{\beta(\theta-1)(s-2)+\theta s+s-2\theta}{2(s-1)}<0\).
Consequently, by (4.8), (4.10), (4.11), and Lemma 4.3, we establish the result of (2.10). □
Proof of Theorem 2.3
We use some similar notation in the proof of Theorem 2.2. Obviously, one has
By the proof of (3.21) of Shen and Xie [1], we establish that
which implies
Since \(\hat{m}_{n}(x)=R_{n}(x)+\tilde{m}_{n}(x)\),
It follows from the proof of (4.8) that
Since \(S_{f}^{\prime}\) is a compact set, there exists a \(\xi>0\) such that \(S_{f}^{\prime}\subset B:=\{x:\|x\|\leq\xi\}\). Let \(v_{n}\) be a positive integer. Take an open covering \(\bigcup_{j=1}^{v_{n}^{d}}B_{jn}\) of B, where \(B_{jn}\subset \{x:\|x-x_{jn}\|\leq\frac{\xi}{v_{n}}\}\), \(j=1,2,\ldots,v_{n}^{d}\), and their interiors are disjoint. So it follows that
By the definition of \(\widetilde{m}_{n}(x)\) in (4.9) and the Lipschitz condition of K,
Taking \(v_{n}=\lfloor\frac{\tau_{n}}{h^{d+1}a_{n}}\rfloor+1\), we obtain
and
In view of \(|E\widetilde{m}_{n}(x_{jn})-E\widetilde{m}_{n}(x)|\leq E|\widetilde{m}_{n}(x_{jn})-\widetilde{m}_{n}(x)|\), we have by (4.17)
For \(1\leq i\leq n\) and \(1\leq j\leq v_{n}^{d}\), denote \(\tilde{Z}_{i}(j)=\frac{Y_{i}K_{h}(x_{jn}-X_{i})}{f(X_{i})}I(|Y_{i}|\leq \tau_{n})\). Then similar to the proof of (4.11), we obtain by Lemma 4.2 with \(m=a_{n}^{-1}\tau_{n}^{-1}\) and \(\varepsilon=Mna_{n}\) for n sufficiently large
where the value of M will be given in (4.22).
In view of \(0<\theta<1\), \(s>2\), \(h=n^{-\theta/d}\), and \(a_{n}=(\frac{\ln n}{nh^{d}})^{1/2}\), one has \(h^{-d(d+1)}=n^{\theta(d+1)}\) and \(a_{n}^{-\frac{sd}{s-1}}=(\ln n)^{-\frac{sd}{2(s-1)}}n^{\frac{sd(1-\theta)}{2(s-1)}}\). Therefore, by \(v_{n}=\lfloor\frac{\tau_{n}}{h^{d+1}a_{n}}\rfloor+1\) and \(\tau_{n}=a_{n}^{-\frac{1}{s-1}}\), we obtain for n sufficiently large
where M is sufficiently large such that
Meanwhile, by (2.11) and \(h=n^{-\theta/d}\), one has for n sufficiently large
in which is used the fact that \(s>2\), \(0<\theta<1\),
and
Thus, by (4.20)-(4.23), we establish that
Finally, the result of (2.12) follows from (4.12)-(4.16), (4.18), (4.19), and (4.24) immediately. □
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Acknowledgements
The authors are deeply grateful to the editors and the anonymous referees for their careful reading and insightful comments, which helped in improving the earlier version of this paper. This work is supported by the National Natural Science Foundation of China (11171001, 11426032, 11501005), Natural Science Foundation of Anhui Province (1408085QA02, 1508085QA01, 1508085J06, 1608085QA02), the Provincial Natural Science Research Project of Anhui Colleges (KJ2014A010, KJ2014A020, KJ2015A065), Quality Engineering Project of Anhui Province (2015jyxm054), Applied Teaching Model Curriculum of Anhui University (XJYYKC1401, ZLTS2015053) and Doctoral Research Start-up Funds Projects of Anhui University.
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Li, X., Yang, W. & Hu, S. Uniform convergence of estimator for nonparametric regression with dependent data. J Inequal Appl 2016, 142 (2016). https://doi.org/10.1186/s13660-016-1087-z
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DOI: https://doi.org/10.1186/s13660-016-1087-z