Uniform in bandwidth consistency of kernel-type function estimators

Abstract

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya–Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators. 1. Introduction and statements of main results. Let X,X1,X2,... be i.i.d. Rd, d ≥ 1, valued random variables and assume that the common distribution function of these variables has a Lebesgue density function, which we shall denote by f. A kernel K will be any measurable function whic