Rates of convergence for partial mass problems

Abstract

We consider a class of partial mass problems in which a fraction of the mass of a probability measure is allowed to be changed (trimmed) to maximize fit to a given pattern. This includes the problem of optimal partial transportation of mass, where a part of the mass need not be transported, and also trimming procedures which are often used in statistical data analysis to discard outliers in a sample (the data with lowest agreement to a certain pattern). This results in a modified, trimmed version of the original probability which is closer to the pattern. We focus on the case of the empirical measure and analyze to what extent its optimally trimmed version is closer to the true random generator in terms of rates of convergence. We deal with probabilities on \({\mathbb{R}^k}\) and measure agreement through probability metrics. Our choices include transportation cost metrics, associated to optimal partial transportation, and the Kolmogorov distance. We show that partial transportation (as opposed to classical, complete transportation) results in a sharp decrease of costs only in low dimension. In contrast, for the Kolmogorov metric this decrease is seen in any dimension.