Wavelet Estimation Of Nonparametric Regression Function Under Dependent Sample

Reviewing the history of regression analysis, the research had focused on parameter regression before the 70's in the twentieth century, while thereafter the research of nonparametric regression was flourished gradually which attracted many statisticians attention, Parameter regression model offers multitudes of extra information for regression function, so it drives people to seek other ways to do the research. Consequently, it caused the research of nonparametric regression, The characteristics of it are arbitrary regression function form and less restriction to the distribution of the variables. Therefore, it is very practicle. Fan.Y.(1990) considered the nonparametric regression model:Y_i⁽ⁿ⁾ = g(t_i⁽ⁿ⁾) +ε_i⁽ⁿ⁾ , i = 1,2,…,n, where {t_i⁽ⁿ⁾}were known and fixed design points, {ε_i⁽ⁿ⁾} were assumed to be both dependent and non-identically distributed random variables. He got the estimation of g(t) with weight function method and discussed its weak, mean square error, and universal consistent under very general conditions on the temporal dependence and heterogeneity ofε_i⁽ⁿ⁾s. Asymptotic distribution of the estimator was also considered. Professor Chai Genxiang considered the nonparametric regression model: Y_i=g(t_i) +ε_i, i=1,2,…,n,where{t_i} were fixed design points and{ε_i} wereρ-depending r.v. andα-depending r.v., He got the estimation of g(t) with wavelet method and discussed its weak convergence, strong convergence and the rate of convergence.In this article the authors mainly consider the nonparametric regression model: Y_ni=g(t_ni) +ε_ni, i = 1,2,…,n, where {t_ni}be a set of fixed designed points and {ε_ni}be stationary process under the two dependent samples:(1)When {ε_ni} is martingale deference sequence, we adopt wavelet method to estimate regression function g(t) and to study its consistence and the rate of consistence.(2) When {ε_ni} is L^q-mixingale, we adopt wavelet method to estimate regression function g(t) and to study its consistence and strong convergence.