Abstract
In this paper, we consider a nonparametric regression model with replicated observations based on the φmixing and the ρmixing error’s structures respectively, for exhibiting dependence among the units. The wavelet procedures are developed to estimate the regression function. Under suitable conditions, we obtain expansions for the bias and the variance of wavelet estimator, prove the moment consistency, the strong consistency, the strong convergence rate of it, and establish the asymptotic normality of wavelet estimator.
Keywords:
nonparametric regression model; wavelet; weak dependence; φmixing; ρmixing1 Introduction
Consider the following nonparametric regression model:
from a discrete set of observations of the process
It is well known that regression model has a wide range of applications in filtering and prediction in communications and control systems, pattern recognition, classification and econometrics, and is an important tool of data analysis. Of much interest about the problem have been the weighted function estimates of g; see, for example, Priestley and Chao [1], Gasser and Müller [2,3], Prakasa Rao [4], Clark [5] and the references therein for the independent case; Roussas [6], Fan [7], Roussas and Tran [8], Liang and Jing [9], Yang and Li [10], Yang [11] for the various dependent cases.
In this article, we discuss a nonparametric estimation problem in the model (1.1)
with repeated measurements. We assume that a random sample of m experimental units is available and the observed data for the jth unit are the values,
where the
In the paper, we develop wavelet methods to estimate a regression function in the
model (1.2) with the φmixing and ρmixing error’s structures respectively, that is,
For a systematic discussion of wavelets and their applications in statistics, see the recent monographs by Härdle et al.[14] and Vidakovic [15]. Due to their ability to adapt to local features of unknown curves, many authors have applied wavelet procedures to estimate the general nonparametric model. See recent works, for example, Antoniadis et al.[16] and Xue [17] on independent errors; Johnstone and Silverman [18] for correlated noise; Liang et al.[19] on martingale difference errors; Li and Xiao [20] for long memory data; Li et al.[21] on associated samples; Xue [22], Sun and Chai [23], Li and Guo [24] and Liang [25] on mixing error assumptions.
For dealing with weakly dependent data, bootstrap and blockwise are well known. They
are useful techniques of resampling, which can preserve the dependent properties of
the data by appropriately choosing blocks of data. They have been sufficiently investigated
by many papers, for example, Bühlman and Künsch [26], Yuichi [27], Lahiri [28], Lin and Zhang [29,30] and Lin et al.[31]. For the nonparametric regression model without repeated observations under weakly
dependent processes, Lin and Zhang [30] respectively adopted bootstrap wavelet and blockwise bootstrap wavelet to generate
an independent blockwise sample from the original dependent data, defined the wavelet
estimators of
Recall the definitions of the sequences of the φmixing and ρmixing random variables. Let
A sequence of random variables
A sequence of random variables
The concept of a mixing sequence is central in many areas of economics, finance and other sciences. A mixing time series can be viewed as a sequence of random variables for which the past and distant future are asymptotically independent. A number of limit theorems for φmixing and ρmixing random variables have been studied by many authors. For example, see Shao [32], Peligrad [33], Utev [34], Kiesel [35], Chen et al.[36] and Zhou [37] for φmixing; Peligrad [38], Peligrad and Shao [39,40], Shao [41] and Bradley [42] for ρmixing. Some limit theories can be found in the monograph of Lin and Lu [43].
The article is structured as follows. In Section 2, we introduce the wavelet estimation procedures and establish main results. The proofs of the main results are provided in Section 3.
2 Estimators and main results
Defining
where
For convenience, we introduce some symbols and definitions along the line of Antoniadis
et al.[16]. Suppose that
where Z denotes the set of integers,
In the sequel, let
Before formulating the main results, we first give some assumptions, which are quite mild and can be easily satisfied.
A1.
(i)
(ii)
(iii)
A2.
(i)
(ii)
(iii)
(iv)
A3.
A4.
A5.
A6.
A7.
A8.
Remark 2.1 We refer to the monograph of Doukhan [44] for properties of φmixing and ρmixing, and more mixing conditions.
Remark 2.2 It is well known that (A4)(A7) are the mild regularity conditions for wavelet smoothing; see Antoniadis et al.[16], Chai and Xu [45], Xue [22], Sun and Chai [23], Zhou and You [46] and Li and Guo [24].
Remark 2.3 (A8) is satisfied easily. For example,
Our results are listed as follows.
Theorem 2.1Assume that (A1)(i) and (A2)(i), and (A4)(A7) are satisfied. Then
(a)
(b) φmixing:
ρmixing:
Theorem 2.2Under (A1)(i), (A2)(i)(ii), and (A4)(A7), we have
Theorem 2.3Assume that [(A1)(ii) and (A2)(i)(ii)] or [(A1)(i) and (A2)(iii)], and (A4)(A7) are satisfied. Then
Theorem 2.4Assume that (A1)(ii), (A2)(i)(ii), and (A4)(A7) are satisfied. If
Theorem 2.5Assume that (A1)(iii), (A2)(i)(iv), (A3), and (A4)(A8) are satisfied. For a fixedxand each
where
3 Proofs of the main results
In order to prove the main results, we first present several lemmas.
Lemma 3.1Suppose that (A6) holds. We have
(a)
(b)
(c)
The proofs of (a) and (b), and (c) respectively can be found in Antoniadis et al.[16] and Walter [47].
Lemma 3.2Suppose that (A6)(A7) hold, and
where
It follows easily from Theorem 3.2 of Antoniadis et al. [16].
Lemma 3.3 (a) Let
(b) Let
Lemmas 3.3(a) and (b) respectively come from Lemmas 10.1.d and 10.1.c of Lin and Lu[43].
Let
Lemma 3.4 (a) Let
(i) If
(ii) Suppose that there exists an array
(b) Let
Lemma 3.5Let
for some
Lemma 3.5 can be found in Theorem 8.2.2 of Lin and Lu [43]. Ibragimov [48,49] gave the following Lemma 3.6, which also can be found in Lin and Lu [43].
Lemma 3.6Let
We are now in a position to give the proofs of the main results.
Proof of Theorem 2.1 From (1.2) and (2.2), we have
By Lemma 3.2, (a) holds.
Denote
For φmixing, by Lemmas 3.3(a) and (3.2), we have
Therefore, (b) holds for φmixing. Similar to the arguments, we obtain (b) for ρmixing by Lemma 3.3(b). □
Proof of Theorem 2.2 We know that
Proof of Theorem 2.3 From (1.2) and (2.2), we have
where
as
(1) φmixing. Here, we consider
If the assumptions are [(A1)(ii) and (A2)(i)(ii)], denote
For
Therefore, it follows from the BorelCantelli lemma that
Note that
By (3.8) and Lemma 3.1(ii), one gets
Further, we have
From (3.6), (3.7), (3.9) and (3.10), we obtain (3.5).
If the assumptions are [(A1)(i) and (A2)(iii)], note that
Therefore, from the BorelCantelli lemma, we obtain (3.5).
(2) ρmixing. We also consider
Note that
If the assumptions are [(A1)(ii) and (A2)(i)(ii)], from (3.6)(3.10), it is known
that we only need to prove (3.7) for obtaining (3.5). Taking
Therefore, (3.7) holds.
If the assumptions are [(A1)(i) and (A2)(iii)], take
Thus, we obtain (3.5).
So, we complete the proof of Theorem 2.3. □
Proof of Theorem 2.4 Here, we use some symbols of the proof of Theorem 2.3. From (3.3), we have
By Lemma 3.2, for
Note that
Note that
Therefore,
By (3.14), one gets
Thus, we obtain
By Lemma 3.1(ii), (3.14),
and
Therefore,
To complete the proof of the theorem, it is suffices to show that
by (3.11)(3.13) and (3.15)(3.16).
Here, we show (3.17) under φmixing and ρmixing, respectively.
(1) φmixing. Taking
Thus, (3.17) holds by the BorelCantelli lemma.
(2) ρmixing. Similar to the arguments in the proof of Theorem 2.3,
Therefore, we also obtain (3.17).
So, we complete the proof of Theorem 2.4. □
Proof of Theorem 2.5 Denote
Since we have
by Lemma 3.2 and (A8), it suffices to show that
and
Let
Therefore,
which can be made arbitrarily small if we first choose ς such that
It remains to establish (3.19). Denote
For any
Hence,
We easily obtain
since
Thus, we complete the proof of Theorem 2.5. □
4 Conclusion and discussion
The paper studies a nonparametric regression model with replicated observations under
weakly dependent processes by wavelet procedures. For exhibiting dependence among
the units, we assume that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The three authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
The authors thank the associate editor and two anonymous referees, whose valuable comments greatly improved the paper. This work is partially supported by Key Natural Science Foundation of Higher Education Institutions of Anhui Province (KJ2012A270), NSFC (11171065, 11061002), Youth Foundation for Humanities and Social Sciences Project from Ministry of Education of China (11YJC790311), NSFJS (BK2011058), National Natural Science Foundation of Guangxi (2011GXNSFA018126) and Postdoctoral Research Program of Jiangsu Province of China (1202013C).
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