Consistency for wavelet estimator in nonparametric regression model with extended negatively dependent samples
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Consistency for wavelet estimator in nonparametric regression model with extended negatively dependent samples Liwang Ding1,2 · Ping Chen1 · Yongming Li3 Received: 28 September 2017 / Revised: 11 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract In this article, we mainly study the consistency properties of wavelet estimator in nonparametric regression model with extended negatively dependent samples. Under some suitable conditions, the pth mean consistency, complete consistency and complete consistency rates of the wavelet estimator in nonparametric regression model with extended negatively dependent samples are obtained. Our results generalize or improve the corresponding ones and the wavelet estimator method for independent and mixing dependent samples to some extent. Keywords Extended negatively dependent samples · Wavelet estimator · Consistency · Regression model Mathematics Subject Classification 62G05 · 62G08
1 Introduction Consider the estimation of a standard nonparametric regression model involving a regression function g(·) which is defined on [0, 1]:
This research is supported by the National Science Foundation of China (11271189, 11461057), the National Science Foundation of Jiangxi (20161BAB201003), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX18_0372), 2017 Youth Teacher Research and Development Fund Project of Guangxi University of Finance and Economics (2017QNA01).
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Liwang Ding [email protected]
1
School of Science, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China
2
School of Information and Statistics, Guangxi University of Finance and Economics, Nanning 530003, People’s Republic of China
3
School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai 200433, People’s Republic of China
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L. Ding et al.
Yni = g(tni ) + εni , (1 ≤ i ≤ n),
(1.1)
where tni are nonrandom design points, tni ’s are denoted t(ni) and taken to be ordered 0 ≤ t(n1) ≤ · · · ≤ t(nn) ≤ 1, εni are random errors. For each n ≥ 1, assume that (εn1 , εn2 , . . . , εnn ) have the same distribution as (ε1 , ε2 , . . . , εn ). It is generally known that nonparametric regression model (1.1) has many applications in practical problems, so model (1.1) and its special cases have been studied extensively by many authors. For example, when the errors εni are independent samples, various estimation methods have been used to derive estimators of g(·). For instance, Clark (1977), Georgiev and Greblicki (1986), Qin (1989), and Xue (2002). Under mixing dependent samples, Yang and Wang (1999) investigated strong consistency for the estimator of model (1.1) under NA samples. Sun and Chai (2004) investigated consistency properties for the estimator of model (1.1) under strong mixing samples. Liang and Jing (2005) investigated asymptotic properties for the estimator of model (1.1) under NA samples. Yang et al. (2012) investigated some consistency results for the estimator of model (1.1) un
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