On the maximal deviation of the kernel regression function estimate

Abstract

Let (X,Y) be a twodimensional random vector and let be a random sample drawn from its distribution. In this paper we consider the kernel estimate r _n (t) of the regression function r(t = E(Y ‖ X = t)that is defined by

where k is a weight function satisfin certain roerties and {a _n } is a seouence of positive numbers tending to zero n → ∞.

With the help of the in variance principle for the empirical process and some consistency statements a limit theorem for the maximum of the normalized deviation of the estimate from the regression function itself is proved.

Keywords:

• kernel estimates for regression functions
• global measure of deviation
• asymptotic distribution
• strong in variance principle for the empirical process