Asymptotic properties of wavelet estimators in semiparametric regression models under dependent errors

Consider the semiparametric regression model y"i=x"i^[email protected]+g(t"i)[email protected]"i for i=1,...,n, where x"[email protected]?R^p are the random design vectors, t"i are the constant sequences on [0,1], @[email protected]?R^p is an unknown vector of the slop parameter, g is an unknown real-valued function defined on the closed interval [0,1], and the error random variables @e"i are coming from a stationary stochastic process, satisfying the strong mixing condition in some results. Under suitable conditions, we obtain expansions for the bias and the variance of wavelet estimators @[email protected]?"n and [email protected]?"n(@?) of @b and g(@?) respectively, prove their weak consistency, and establish the asymptotic normality and the Berry-Esseen bound of @[email protected]?"n.

[1]  J. Rice.  Convergence rates for partially splined models , 1986 .

[2]  G. Walter.  Wavelets and other orthogonal systems with applications , 1994 .

[3]  Young K. Truong,et al.  Local Linear Estimation in Partly Linear Models , 1997 .

[4]  Han-Ying Liang,et al.  On a semiparametric regression model whose errors form a linear process with negatively associated innovations , 2006 .

[5]  Du Xue-qiao,et al.  Nonparametric wavelet estimator of a fixed designed regression function for martingale sequences , 2007 .

[6]  P. Doukhan.  Mixing: Properties and Examples , 1994 .

[7]  Wang Xiaozhi,et al.  Convergence Rate of Wavelet Estimator in Semiparametric Models with Dependent MA(1) Error Process , 2010 .

[9]  Han-Ying Liang,et al.  Asymptotics of estimators in semi-parametric model under NA samples , 2006 .

[10]  G Chai,et al.  Wavelet Smoothing in Semiparametric Regression Model , 1999 .

[11]  Shanchao Yang,et al.  Maximal Moment Inequality for Partial Sums of Strong Mixing Sequences and Application , 2007 .

[12]  李林元,et al.  Wavelet-based estimators of mean regression function with long memory data , 2006 .

[13]  V. Volkonskii,et al.  Some Limit Theorems for Random Functions. II , 1959 .

[14]  Yang Shan-chao,et al.  Uniformly Asymptotic Normality of the Regression Weighted Estimator for -mixing Samples , 2006 .

[15]  Han-Ying Liang,et al.  Asymptotic normality in partial linear models based on dependent errors , 2009 .

[16]  P. Speckman.  Kernel smoothing in partial linear models , 1988 .

[17]  P. Whittle,et al.  Bounds for the Moments of Linear and Quadratic Forms in Independent Variables , 1960 .

[18]  Han-Ying Liang,et al.  ASYMPTOTIC NORMALITY OF WAVELET ESTIMATOR OF REGRESSION FUNCTION UNDER NA ASSUMPTIONS , 2007 .

[19]  Yong Zhou,et al.  Consistency and uniformly asymptotic normality of wavelet estimator in regression model with associated samples , 2008 .

[20]  Hung Chen,et al.  Convergence Rates for Parametric Components in a Partly Linear Model , 1988 .

[21]  Z. Q. John Lu,et al.  Nonlinear Time Series: Nonparametric and Parametric Methods , 2004, Technometrics.

[22]  Lorraine Denby.  Smooth Regression Functions (partial Residual Plots). , 1984 .

[23]  Han-Ying Liang,et al.  A BERRY-ESSEEN TYPE BOUND OF REGRESSION ESTIMATOR BASED ON LINEAR PROCESS ERRORS , 2008 .

[24]  Wolfgang Härdle,et al.  Partially Linear Models , 2000 .

[25]  George G. Roussas,et al.  Fixed design regression for time series: asymptotic normality , 1992 .

[26]  Haiyan Cai,et al.  Wavelet change‐point estimation for long memory non‐parametric random design models , 2010 .

[27]  Zhengyan Lin,et al.  Limit Theory for Mixing Dependent Random Variables , 1997 .

[28]  Nouna Kettaneh,et al.  Statistical Modeling by Wavelets , 1999, Technometrics.

[29]  A. A. Weiss,et al.  Semiparametric estimates of the relation between weather and electricity sales , 1986 .

[30]  Chengdong Wei,et al.  Berry-Esseen bounds for wavelet estimator in a regression model with linear process errors , 2011 .

[31]  H. Hongchang,et al.  Wavelet Estimation in Semiparametric Regression Models With Martingale Differences Time Series Errors , 2010 .

[32]  Han-Ying Liang,et al.  Asymptotic normality of wavelet estimator in heteroscedastic model with α-mixing errors , 2011, J. Syst. Sci. Complex..

[33]  Wei Liu,et al.  Convergence Rate of Wavelet Estimator in a Semi-parametric Regression Model with alpha-mixing Errors , 2011, 2011 International Conference on Computational and Information Sciences.

[34]  Hongchang Hu,et al.  Convergence rates of wavelet estimators in semiparametric regression models under NA samples , 2012 .

[35]  Sara van de Geer,et al.  Penalized quasi-likelihood estimation in partial linear models , 1997 .

[36]  A. Antoniadis,et al.  Wavelet Methods for Curve Estimation , 1994 .

[37]  Yongming Li,et al.  Asymptotic normality of wavelet estimator for strong mixing errors , 2009 .

[38]  Dongxiao Zhang,et al.  Wavelet estimation in nonparametric model under martingale difference errors , 2004 .

[39]  Xian Zhou,et al.  Wavelet estimation in varying-coefficient partially linear regression models , 2004 .

[40]  Nancy E. Heckman,et al.  Spline Smoothing in a Partly Linear Model , 1986 .

[41]  D Hu.  Weighted Wavelet Estimation in Semiparametric Regression Models with ψ-mixing Heteroscedastic Errors , 2013 .

[42]  Xue Ll.  RATES OF RANDOM WEIGHTING APPROXIMATION OF WAVELET ESTIMATES IN SEMIPARAMETRIC REGRESSION MODEL , 2003 .

[43]  Germán Aneiros,et al.  Asymptotic properties in partial linear models under dependence , 2001 .

[44]  Ling Neng-xiang.  Strong consistency of wavelet estimation in the semiparametric regression model , 2009 .