Uniform in bandwidth exact rates for a class of kernel estimators

Abstract

Given an i.i.d sample $(Y_i,Z_i)$, taking values in $\RRR^{d'}\times \RRR^d$, we consider a collection Nadarya-Watson kernel estimators of the conditional expectations $\EEE(<c_g(z),g(Y)>+d_g(z)\mid Z=z)$, where $z$ belongs to a compact set $H\subset \RRR^d$, $g$ a Borel function on $\RRR^{d'}$ and $c_g(\cdot),d_g(\cdot)$ are continuous functions on $\RRR^d$. Given two bandwidth sequences $h_n<\wth_n$ fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in $g\in\GG,\;z\in H$ and $h_n\le h\le \wth_n$ under mild conditions on the density $f_Z$, the class $\GG$, the kernel $K$ and the functions $c_g(\cdot),d_g(\cdot)$. We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities $\PPP(Y\in C\mid Z=z)$, that hold uniformly in $z\in H,\; C\in \CC,\; h\in [h_n,\wth_n]$. Here $\CC$ is a Vapnik-Chervonenkis class of sets.