Rates of convergence for partial mass problems
type: Item
instance of: scholarly article
title: Rates of convergence for partial mass problems
zbMATH Open document ID: 1286.60004
author: Eustasio Del Barrio
author: Carlos Matrán
publication date: 2013-05-13T00:00:00Z
review text: Given two probability measures \(p\) and \(R\) on \(\mathbb{R}^k\) and \(\alpha\in [0,1]\), \(R\) is said to be a trimming of \(P\) if it is absolutely continuous with respect to \(P\) and the Radon-Nikodym derivative satisfies \(dR/dP\leq 1/(1-\alpha)\). The set of \(\alpha\)-trimmings of \(P\) is denoted by \(\mathcal{R}_{\alpha}(P)\). The authors consider the minimisation problem \[ \inf_{R\in\mathcal{R}_{\alpha}(P)} d(R,P_0), \] where \(P_0\) is a reference probability and \(d\) is either the Kolmogorov distance \(d_K\) or the \(L^p\)-Wasserstein distance \(W_p\). If they exist, the following probabilities are considered \[ P_{n,\alpha}:=\arg\,\min_{\mathcal{R}_{\alpha}(P_n)} d(R,P_0)\quad\text{and}\quad P_{\alpha}:=\arg\,\min_{\mathcal{R}_{\alpha}(P)} d(R,P_0), \] where \(P_n\) is the empirical measure on a sample. The authors obtain rates of convergence for \(W_p\): \[ W_p(\mathcal{R}_{\alpha}(P_n),P)=o_P\left(\frac{(\log n)^{\nu}}{n^{1/k}}\right), \quad \nu>1/k, \] and, for \(d_K\): \[ d_k(\mathcal{R}_{\alpha}(P_n),P)=o_P\left(\frac{(\log n)^{\nu}}{n}\right)^{\frac{k+1}{2k}}. \]
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60A10
zbMATH DE Number: 6162387
zbMATH Keywords: partial mass transportation problem
zbMATH Keywords: random quantization
zbMATH Keywords: optimal transportation plan
zbMATH Keywords: similarity
zbMATH Keywords: trimming
zbMATH Keywords: trimmed probability
zbMATH Keywords: Kolmogorov distance
zbMATH Keywords: Wasserstein distance
zbMATH Keywords: rate of convergence
zbMATH Keywords: concentration of measure
reviewed by: Carlo Sempi
MaRDI profile type: Publication
full work available at URL: s00440-011-0406-z
OpenAlex ID: W2067018176
cites work: On optimal matchings
cites work: Trimmed Comparison of Distributions
cites work: Uniqueness and approximate computation of optimal incomplete transportation plans
cites work: Similarity of samples and trimming
cites work: Central limit theorems for the Wasserstein distance between the empirical and the true distributions
cites work: Asymptotics for \(L_2\) functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances
cites work: Some asymptotic theory for the bootstrap
cites work: Concentration inequalities using the entropy method
cites work: Free boundaries in optimal transport and Monge-Ampère obstacle problems
cites work: Limit theorems for random normalized distortion
cites work: Q5288953
cites work: Combinatorial methods in density estimation
cites work: Asymptotics for transportation cost in high dimensions
cites work: Asymptotic Minimax Character of the Sample Distribution Function and of the Classical Multinomial Estimator
cites work: The optimal partial transport problem
cites work: Best approximations to random variables based on trimming procedures
cites work: Foundations of quantization for probability distributions
cites work: Mean rates of convergence of empirical measures in the Wasserstein metric
cites work: On large deviations of the empiric D.F. of vector chance variables and a law of the iterated logarithm
cites work: On the Deviations of the Empiric Distribution Function of Vector Chance Variables
cites work: Q3915689
cites work: Multidimensional Lévy inequalities and their applications
cites work: Concentration inequalities and model selection. Ecole d'Eté de Probabilités de Saint-Flour XXXIII -- 2003.
cites work: Convergence of stochastic processes
cites work: Limit theorems for multi-dimensional random quantizers
cites work: On the Kolmogorov and Smirnov Limit Theorems for Discontinuous Distribution Functions
cites work: Matching random samples in many dimensions
cites work: The integrability of the square exponential transportation cost
cites work: The transportation cost from the uniform measure to the empirical measure in dimension \(\geq 3\)
cites work: Weak convergence and empirical processes. With applications to statistics
cites work: Asymptotic quantization error of continuous signals and the quantization dimension
DOI: 10.1007/S00440-011-0406-Z
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 6162387