Consistent nonparametric multiple regression: the fixed design case
Consider the nonparametric regression model \(Y_ i^{(n)}=g(x_ i^{(n)})+\epsilon_ i^{(n)}\), \(i=1,...,n\), where \(g\) is an unknown function, the design points \(x_ i^{(n)}\) are known and nonrandom, and \(\epsilon_ i^{(n)}\)'s are independent random variables. The regressor is assumed to take values in \(A\subset \mathbb R^ p\), and the regressand to be real valued. This paper studies the behavior of the general nonparametric estimate \[ g_ n(x)=\sum^{n}_{i=1}w_{ni}(x)Y_ i^{(n)}, \] where the weight function \(w_{ni}\) is of the form \(w_{ni}(x)=w_{ni}(x;x_ i^{(n)},\dots,x_ n^{(n)})\). Under suitable conditions, it is shown that the general linear smoother \(g_ n\) for the unknown rendent variable.
scientific article; zbMATH DE number 4039029
scientific article