Uniform in bandwidth exact rates for a class of kernel estimators

2012/01/26 by Davit Varron, Varron, Davit, Ingrid Van Keilegom +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1201.5507

Published in the Annals of the Institute of Statistical Mathematics Volume 63, p. 1077-1102 (2011)

arxiv created 2012/01/26 · openalex publication_date 2012/01/26 · arxiv updated 2012/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an i.i.d sample (Y_i,Z_i), taking values in \RRR^d'× \RRR^d, we consider a collection Nadarya-Watson kernel estimators of the conditional expectations \EEE(<c_g(z),g(Y)>+d_g(z)| Z=z), where z belongs to a compact set H⊂ \RRR^d, g a Borel function on \RRR^d' and c_g(⋅),d_g(⋅) 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∈\GG, z∈ H and h_n≤ h≤ \wth_n under mild conditions on the density f_Z, the class \GG, the kernel K and the functions c_g(⋅),d_g(⋅). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \PPP(Y∈ C| Z=z), that hold uniformly in z∈ H, C∈ \CC, h∈ [h_n,\wth_n]. Here \CC is a Vapnik-Chervonenkis class of sets.

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