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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 (n) =g(x i (n) )+ϵ i (n) ,i=1,···,n,

where g is an unknown regression function and assumed to be bounded and real valued on A p , x i (n) ’s are known and fixed design points and ϵ i (n) ’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 (x)= i=1 n W ni (x)Y i (n) ,

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

62G05Nonparametric estimation
62E20Asymptotic distribution theory in statistics
60G44Martingales with continuous parameter
60F05Central limit and other weak theorems