Convergence of the empirical process in Mallows distance, with an application to bootstrap performance

2004/06/29 by Richard J. Samworth, Richard Samworth, Samworth, Richard +2 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #60F25 #62E20 #62F40 #Bayesian Methods and Mixture Models #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST) #math.PR #math.ST #msc:60F25 #msc:62E20 #msc:62F40 #stat.TH

paper · pdf · doi:10.48550/arxiv.math/0406603

18 pages, 1 figure

arxiv created 2004/06/29 · openalex publication_date 2004/06/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the rate of convergence of the Mallows distance between the empirical distribution of a sample and the underlying population. The surprising feature of our results is that the convergence rate is slower in the discrete case than in the absolutely continuous setting. We show how the hazard function plays a significant role in these calculations. As an application, we recall that the quantity studied provides an upper bound on the distance between the bootstrap distribution of a sample mean and its true sampling distribution. Moreover, the convenient properties of the Mallows metric yield a straightforward lower bound, and therefore a relatively precise description of the asymptotic performance of the bootstrap in this problem.

Citations

• Bootstrap Methods: Another Look at the Jackknife

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• Distributional Limit Theory for Optimal Transport

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