Uniform in bandwidth exact rates for a class of kernel estimators

TL;DR

This paper establishes precise uniform almost sure bounds for a class of kernel estimators of conditional expectations, enabling the construction of reliable confidence intervals for conditional probabilities with explicit rates.

Contribution

It provides explicit, uniform almost sure deviation bounds for kernel estimators over a range of bandwidths, extending their applicability to confidence interval construction.

Findings

Derived exact almost sure limit bounds for kernel estimator deviations.

Applied results to build uniform confidence intervals for conditional probabilities.

Demonstrated the use of smoothed empirical likelihood in nonparametric inference.

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.