On Almost Sure Convergence of Conditional Empirical Distribution Functions

Abstract

We investigate the almost sure convergence of a kernel-type conditional empirical distribution function both in sup-norm and weighted sup-norms. As an application we get a strong law for the Nadaraya-Watson estimate of a regression function $m(\mathbf{x}) = \mathbb{E}(Y\mid \mathbf{X} = \mathbf{x})$ under a weak moment condition on $Y$.