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The Berry-Esséen bounds for kernel density estimator under dependent sample
Journal of Inequalities and Applications volume 2012, Article number: 287 (2012)
Let be a φ-mixing sequence with an unknown common probability density function and the mixing coefficients satisfy . By using some inequalities for φ-mixing random variables and selecting some positive bandwidths , we investigate the Berry-Esséen bounds of the estimator for and its bounds are presented as and , where .
The most popular nonparametric estimator of a distribution based on a sample of observations is the empirical distribution, and the most popular method of nonparametric density estimation is the kernel method. For an introduction and applications of this field, the books by Prakasa Rao  and Silverman  provide the basic methods for density estimation. For the nonparametric curve estimation from time series such as φ-mixing, ρ-mixing and α-mixing, Györfi et al.  studied the density estimator and hazard function estimator for these mixing sequences. It is known that φ-mixing ⇒ ρ-mixing ⇒ α-mixing, and its converse is not true. Although, φ-mixing is stronger than α-mixing, some properties of φ-mixing such as moment inequality, exponential inequality, etc., are better than those of α-mixing to use. For the properties and examples of mixing, we can read the book of Doukhan . In this paper, we only give the definition of a φ-mixing sequence. For the basic properties of φ-mixing, one can refer to Billingsley .
Denote and define the coefficients as follows:
If as , then is said to be a φ-mixing sequence.
Many works have been done for the kernel density estimation. For example, Masry  gave the recursive probability density estimation under a mixing-dependent sample, Fan and Yao  summarized the nonparametric and parametric methods including a nonparametric density estimator for nonlinear time series such as φ-mixing, α-mixing, etc. For an independent sample, Cao  investigated the bootstrap approximations in a nonparametric density estimator and obtained Berry-Esséen bounds for the kernel density estimation. Under φ-mixing dependence errors, Li et al.  obtained the asymptotic normality of a wavelet estimator of the regression model. Li et al.  also gave the Berry-Esséen bound of a wavelet estimator of the regression model. Meanwhile, Yang et al.  studied the Berry-Esséen bound of sample quantiles for φ-mixing random variables. In this paper, we will investigate the Berry-Esséen bounds for a kernel density estimator under a φ-mixing dependent sample.
Let be a φ-mixing sequence with an unknown common probability density function and the mixing coefficients satisfy . With the help of techniques of inequalities such as moment inequality, exponential inequality and the Bernstein’s big-block and small-block procedure, by selecting some positive bandwidths , which do not depend on the mixing coefficients and the lengths of Bernstein’s big-block and small-block, we investigate the Berry-Esséen bounds of the estimator for and its bounds are presented as and , where . Particularly, if and , the bound is presented as . For details, please see our results in Section 3. Some assumptions and lemmas are presented in Section 2. Regarding the technique of Bernstein’s big-block and small-block procedure, the reader can refer to Masry [6, 12], Fan and Yao , Roussas  and the references therein.
For the kernel density estimator under association and a negatively associated sample, one can refer to Roussas  and Liang and Baek  obtained for asymptotic normality, Wei  for the consistences, Henriques and Oliveira  for exponential rates, Liang and Baek  for the Berry-Esséen bounds, etc. Regarding other works about the Berry-Esséen bounds, we can refer to Chang and Rao  for the Kaplan-Meier estimator, Cai and Roussas  for the smooth estimator of a distribution function, Yang  for the regression weighted estimator, Dedecker and Prieur  for some new dependence coefficients, examples and applications to statistics, Yang et al.  for sample quantiles under negatively associated sample, Herve et al.  for M-estimators of geometrically ergodic Markov chains, and so on. On the other hand, Härdle et al.  summarized the Berry-Esséen bounds of partially linear models (see Chapter 5 of Härdle et al. ).
Throughout the paper, denote some positive constants not depending on n, which may be different in various places, means the largest integer not exceeding x and is the indicator function of the set A. Let be some positive constant depending only on x. For convenience, we denote in this paper, whose value may vary at different places.
2 Some assumptions and lemmas
For the unknown common probability density function , we assume that
where α is a positive constant and is a family of probability density functions having derivatives of s th order, are continuous and , .
Let be a kernel function in R and satisfy the following condition (A1):
(A1) Assume that is a bounded probability density function and , where is a class of functions with the properties
Here A is a finite constant and s is a positive integer for .
Obviously, the probability density functions Gaussian kernel and Epanechnikov kernel () belong to . For more details, one can refer to Chapter 2 of Prakasa Rao .
For a fixed x, the kernel-type estimator of is defined as
where is a sequence of positive bandwidths tending to zero as .
Similar to the proof of Theorem 2.2 of Wei , we have, by using Taylor’s expansion for , that
where . By (2.1) and (2.2), it follows
For , one can get the ‘bias’ term rate as
by providing .
It can be checked that and () belong to . So, with , one can see that satisfies the conditions and as . Consequently, we pay attention to the Berry-Esséen bound of the centered variate as
in this paper.
(A2) Assume that are the joint p.d.f. of the random variables and , , which satisfy
Under the assumption (A2) and other conditions, Masry  gave the asymptotic normality for the density estimator under a mixing dependent sample and Roussas  obtained the asymptotic normality for the kernel density estimator under an association sample. Unlike the mixing case, association and negatively associated random variables are subject to the transformation , , losing in the process the association or negatively associated property, i.e., the kernel weights , , are not necessarily association or negatively associated random variables (see Roussas  and Liang and Baek [14, 17]). In addition, if (), which is a function of bounded variation, then , where () and () are bounded and monotone nonincreasing functions. Although the transformations and are also the association or negatively associated random variables, and are not integrable in R. So, there are some difficulties in investigating the kernel density estimator under these dependent samples. Meanwhile, the nonparametric estimation and nonparametric tests for association and negatively associated random variables can be found in Prakasa Rao .
In order to obtain the Berry-Esséen bounds for the kernel density estimator under a φ-mixing sample, we give some useful inequalities such as covariance inequality, moment inequality, characteristic function inequality and exponential inequality for a φ-mixing sequence.
Lemma 2.1 (Billingsley , inequality (20.28), p.171)
If and (ξ measurable and η measurable ), then
Lemma 2.2 (Yang , Lemma 2)
Let be a mean zero φ-mixing sequence with . Assume that there exists some such that for all . Then
where C is a positive constant depending only on .
Lemma 2.3 (Li et al. , Lemma 3.4)
Let be a φ-mixing sequence. Suppose that p and q are two positive integers. Set for . Then
Lemma 2.4 Let X and Y be random variables. Then for any ,
Lemma 2.5 (Yang et al. , Corollary A.1)
Let be a mean zero φ-mixing sequence with , a.s., for all . For , let and . Then for and ,
where , .
3 Main results
Theorem 3.1 For , let the condition (A1) hold true. Assume that is a sequence of identically distributed φ-mixing random variables with the mixing coefficients . If , as and , then
where is the standard normal distribution function.
Proof It can be found that
where . We employ the Bernstein’s big-block and small-block procedure to prove (3.1). Denote
and . Define , , as follows:
By (3.2), (3.4), (3.5) and (3.6), one has
From (3.5) and (3.7), it follows
We have by (2.1) and (A1) that
So, by the conditions , and , we apply Lemma 2.2 with and obtain that
Meanwhile, one has , , . With ,
but since , , we have, by applying Lemma 2.1 with and (3.3), that
So, by (3.8), (3.9) and (3.10), one has
On the other hand, by , and Lemma 2.1 with , we obtain that
Now, we turn to estimate . Define
Since , one has
Combining (3.11) with (3.12), one can check that
With , , , one has
So, similar to the proof of (3.10), by Lemma 2.1 with , and , we have that
by (3.13), (3.14) and (3.15), we obtain that
Let , , be the independent random variables and have the same distribution as for . Put . It can be seen that
Denote the characteristic functions of and by and , respectively. Using the Esséen inequality (Petrov , Theorem 5.3), for any , we have
It is a simple fact that
and , . Applying Lemma 2.2 with , we obtain by and that
Consequently, by Lemma 2.3, the Jensen inequality, , (3.3), (3.4) and (3.19), one can see that
Combining (3.18) with (3.20), we obtain, by taking , that
From (3.16), it follows . Thus, by the Berry-Esséen inequality (Petrov , Theorem 5.7), (3.3) and (3.19), one has that
By (3.18) and (3.23), take , we obtain that
Therefore, similar to the proof of (2.28) in Yang et al. , by (3.16), one has
and from (3.22), it follows
Consequently, by (3.17), (3.18), (3.21), (3.24), (3.25) and (3.26), one has that
On the other hand, let . By (3.7), we apply Lemma 2.4 with and obtain that
Obviously, by (3.11) and Markov’s inequality, we have
It is time to estimate . By and (3.12), one has
So, we have, by Lemma 2.5 with and , that for n large enough,
Finally, the desired result (3.1) follows from (3.2), (3.7), (3.27), (3.28), (3.29) and (3.30) immediately. □
Theorem 3.2 For , let the conditions (A1) and (A2) hold true. Assume that is a sequence of identically distributed φ-mixing random variables with the mixing coefficients , and satisfies a Lipschitz condition. If , , then for any ,
where with and is the standard normal distribution function.
Proof By the condition (A1), implies that . Thus, by the Lipschitz condition of , we obtain that
Obviously, one has
Thus, we obtain by combining (3.32) with (3.33) that
Meanwhile, for , one has by the condition (A2) that
By (3.35), we take and obtain that
Applying Lemma 2.2 with , and , we obtain that
Consequently, by (3.34), (3.36), (3.37) and (3.38), it can be checked that
We obtain, by (3.2), (3.31) and (3.38), that
From (3.38) and (3.39), it follows , since as and . Thus, by applying Theorem 3.1, we establish that
On the other hand, similar to the proof of (2.34) in Yang et al. , it follows by (3.39) again that
Finally, by (3.40), (3.41) and (3.42), (3.31) holds true. □
Remark 3.1 Under an independent sample, Cao  studied the bootstrap approximations in nonparametric density estimation and obtained Berry-Esséen bounds as and (see Theorem 1 and Theorem 2 of Cao ). Under a negatively associated sample, Liang and Baek  studied the Berry-Esséen bound and obtained the rate under some conditions (see Remark 3.1 of Liang and Baek ). In our Theorem 3.1 and Theorem 3.2, under the mixing coefficients condition and other simple assumptions, we obtain the Berry-Esséen bounds of the centered variate as and , where . Particularly, by taking and in Theorem 3.2, the Berry-Esséen bound of the centered variate is presented as
where and are defined in Theorem 3.2.
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The authors are grateful to associate editor prof. Andrei Volodin and two anonymous referees for their careful reading and insightful comments. This work was supported by the National Natural Science Foundation of China (11171001, 11201001, 11126176), HSSPF of the Ministry of Education of China (10YJA910005), the Natural Science Foundation of Anhui Province (1208085QA03) and the Provincial Natural Science Research Project of Anhui Colleges (KJ2010A005).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Yang, W., Hu, S. The Berry-Esséen bounds for kernel density estimator under dependent sample. J Inequal Appl 2012, 287 (2012). https://doi.org/10.1186/1029-242X-2012-287
- kernel density estimation
- Berry-Esséen bound
- φ-mixing sequence