UNIFORM RATIO LIMIT-THEOREMS FOR EMPIRICAL PROCESSES

Analysis of non-parametric estimators is easier if one makes use of re fined empirical process inequalities. These inequalities have not been widely appreciated or applied, perhaps because they lie buried quite deep in the empirical process literature. This paper seeks to explain a novel empirical process technique that might make the results more a ccessible. A simple method is presented for proving maximal inequaliti es for empirical processes indexed by unbounded classes of functions. The main result is stated in the form of a tail bound for the ratio of a centered empirical measure to a sum of various L(1) norms. The meth od avoids the usual chaining argument. The bound delivers the correct rates of convergence for the sorts of quantities encountered in non-pa rametric smoothing applications.