Empirical Processes in Action: A Review

Abstract

Summary A review is given of recent applications of empirical process theory and methods to statistics with emphasis on empirical processes indexed by sets and functions. After a brief survey of empirical process theory, we review applications of this theory to estimation (censoring, truncation, biased sampling, regression and density function estimation, minimum distance methods), testing (classical goodness of fit and minimum distance tests, permutation and bootstrap tests, local alternatives and power), pattern recognition, clustering, and classification, bootstrapping of empirical measures, and the delta method. One new theorem on the asymptotic behavior of empirical processes under local alternatives is presented. tion; Clustering; Density function; Differentiable functions; Donsker theorem or CLT; Empirical distribution function; Empirical measure; Empirical process; Estimation Glivenko-Cantelli theorem or SLLN; Inequalities; Local alternatives; Minimum distance; Nonstandard asymptotics; Omnibus tests; Pattern recognition; Permutation tests; Regression function; Trunction; U-processes; VapnikChervonenkis classes.