Yale University
Department of Statistics

ON OPTIMAL MATCHING AND BOOTSTRAP OF EMPIRICAL PROCESSES


Monday, October 14, 1996

Yongzhao Shao
Department of Statistics
Columbia University
New York, New York 10027
yshao@stat.columbia.edu
Seminar to be held in Room 107, 24 Hillhouse

Abstract
Matching problems have arisen in a number of interesting and seemingly unrelated areas. In the last decade, there have been major advances in optimal matching problems using probabilistic methods (see e.g., Statistical Science, 1993, No.1). We first show that the problem of bootstrapping empirical measures may be transformed into a problem of optimal matching. Some recently obtained optimal matching results are applied to establish rate of convergence results on bootstrapped empirical measures with respect to the dual bounded Lipschitz metric as well as the Prohorov distance (i.e. Glivenko-Cantelli type theorems) in all dimensions. The connection between matching probelms and Efron's bootstrap is then used through refining a representation method, introduced in Lo (1989), to obtain rate of convergence results for bootstrapping empirical processes exemplified by the empirical process indexed by Vapnik-Cervonenkis classes of sets. For a particular example, the Donsker type central limit theorem does not hold for the uniform empirical process indexed by lower layers in the unit square, however the supremum distance between a version of the empirical process and its bootstrapped counterpart is shown to go to zero almost surely, with a rate faster than ``square-root n.'' The bootstrap results obtained here complement the well-known results of Gin{\'e} and Zinn (1990) on bootstrapping CLT of empirical processes.