Nonparametric regression for locally stationary random fields

摘要

In this study, we consider the following model: Y_(s_j,A_n) =m(s_j/A_n,X_(s_j,A_n))+ε_(s_j,A_n),s_j∈R_n=[0,A_n]~d，j=1,...，n， (1) where E[ε(_s,A_n)∣X_(s,A_n)] = 0 and R_n is a sampling region. Here, Y_(s_j,A_n)) and X_(s,A_n)are random variables of dimension 1 and p, respectively. We assume that {X_(s,A_n) ： s ∈ R_n} is a locally stationary random field on R_n ⊂ R~d (d ≥ 2). The goals of this study are (ⅰ) to derive uniform convergence rate of kernel estimators for the mean function m in the model(1)on compact sets, (ⅱ) to derive asymptotic normality of the estimators at a given point and (ⅲ)give examples of locally stationary random fields on R~d with a detailed discussion on their properties.