Submitted to Scandinavian Journal of Statistics Uniform Ratio Limit Theorems for Empirical Processes

Uniform ratio limit theorems for empirical processes by David Pollard Statistics Department Yale University Box 208290 Yale Station New Haven, CT 06520-8290 Email: [email protected] September 1993 Revised: October 1993 Second revision: January 1995 Abstract. Analysis of nonparametric estimators is easier if one makes use of re ned 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 accessible. A simple method is presented for proving maximal inequalities 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 empirical to a sum of various L1 norms. The method avoids the usual chaining argument. The bound delivers the correct rates of convergence for the sorts of quantities encountered in nonparametric smoothing applications.