Nonparametric estimation of a regression function with dependent observations

This paper investigates performance of nonparametric kernel regression and its associated bandwidth selection for dependent observations. For short range dependent observations, it is shown that the convergence rate of asymptotic normality and the strong uniform convergence rate (SUCR) of the kernel estimator are of the same orders as those given for the case of independent observations. Also, Mallows' criterion is adjusted to correct for the effect of dependence on bandwidth selection. The bandwidth produced by modified Mallows' criterion is analyzed by a central limit theorem. The convergence rate of the bandwidth is of the same order as that given for the case of independent observations. On the other hand, for long range dependent observations, the SUCR of the kernel estimator could be slower or faster than that given for the case of independent observations, depending on the dependence structure.

[1]  Shean-Tsong Chiu,et al.  On the asymptotic distributions of bandwidth estimates , 1990 .

[2]  W. Härdle.  Applied Nonparametric Regression , 1991 .

[3]  Wolfgang Härdle,et al.  Applied Nonparametric Regression: The kernel method , 1990 .

[4]  M. Priestley,et al.  Non‐Parametric Function Fitting , 1972 .

[5]  J. Marron,et al.  Comparison of Two Bandwidth Selectors with Dependent Errors , 1991 .

[6]  W. Härdle,et al.  How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum , 1988 .

[7]  J. Hart.  Kernel regression estimation with time series errors , 1991 .

[8]  H. Müller.  Nonparametric regression analysis of longitudinal data , 1988 .

[9]  Peter Hall,et al.  Nonparametric regression with long-range dependence , 1990 .

[10]  J. Rice.  Bandwidth Choice for Nonparametric Regression , 1984 .

[11]  Shean-Tsong Chiu,et al.  Bandwidth selection for kernel estimate with correlated noise , 1989 .

[12]  U. Stadtmüller,et al.  Asymptotic properties of nonparametric curve estimates , 1986 .

[13]  Young K. Truoung.  Nonparametric curve estimation with time series errors , 1992 .

[14]  J. Hart,et al.  Kernel Regression Estimation Using Repeated Measurements Data , 1986 .

[15]  Pi-Erh Lin,et al.  Nonparametric estimation of a regression function , 1981 .

[16]  M. C. Jones,et al.  Spline Smoothing and Nonparametric Regression. , 1989 .

[17]  Louis H. Y. Chen.  Two central limit problems for dependent random variables , 1978 .

[18]  J. Marron.  Automatic smoothing parameter selection: A survey , 1988 .

[19]  W. Härdle.  Smoothing Techniques: With Implementation in S , 1991 .

[20]  Hans-Georg Müller.  Longitudinal Data and Regression Models , 1988 .

[21]  R. Olshen.  Asymptotic properties of the periodogram of a discrete stationary process , 1967, Journal of Applied Probability.

[22]  Nonparametric Estimation of a Regression Function: Limiting Distribution2 , 1981 .

[23]  Naomi Altman,et al.  Kernel Smoothing of Data with Correlated Errors , 1990 .

[24]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .