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Consistent nonparametric multiple regression for dependent heterogeneous processes: the fixed design case. (English) Zbl 0698.62040

Summary: Consider the nonparametric regression model

${Y}_{i}^{\left(n\right)}=g\left({x}_{i}^{\left(n\right)}\right)+{ϵ}_{i}^{\left(n\right)},\phantom{\rule{1.em}{0ex}}i=1,···,n,$

where $g$ is an unknown regression function and assumed to be bounded and real valued on $A\subset {ℝ}^{p}$, ${x}_{i}^{\left(n\right)}$’s are known and fixed design points and ${ϵ}_{i}^{\left(n\right)}$’s are assumed to be both dependent and non-identically distributed random variables.

This paper investigates the asymptotic properties of the general nonparametric regression estimator

${g}_{n}\left(x\right)=\sum _{i=1}^{n}{W}_{ni}\left(x\right){Y}_{i}^{\left(n\right)},$

where the weight function ${W}_{ni}\left(x\right)$ is of the form ${W}_{ni}\left(x\right)={W}_{ni}\left(x;{x}_{1}^{\left(n\right)},{x}_{2}^{\left(n\right)},\cdots ,{x}_{n}^{\left(n\right)}\right)$. The estimator ${g}_{n}\left(x\right)$ is shown to be weak, mean square error, and universal consistent under very general conditions on the temporal dependence and heterogeneity of ${ϵ}_{i}^{\left(n\right)}$’s. Asymptotic distribution of the estimator is also considered.

MSC:
 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics 60G44 Martingales with continuous parameter 60F05 Central limit and other weak theorems