Strong convergence of kernel estimates of nonparametric regression functions
Let (X,Y), \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) be i.i.d. random vectors taking values in \(R_ d\times R\) with \(E(| Y|)<\infty\). To estimate the regression function \(m(x)=E(Y| X=x)\), we use the kernel estimate \[ m_ n(x)=\sum^{n}_{i=1}K((X_ i-x)/h_ n)Y_ i/\sum^{n}_{j=1}K((X_ j-x)/h_ n), \] where K(x) is a kernel function and \(h_ n\) a window width. In this paper, we establish the strong consistency of \(m_ n(x)\) when \(E(| Y|^ p)<\infty\) for some \(p>1\) or \(E\{\exp (t| Y|^{\lambda})\}<\infty\) for some \(\lambda >0\) and \(t>0\). It is remarkable that other conditions imposed here are independent of the distribution of (X,Y).
scientific article; zbMATH DE number 3932201