Best finite constrained approximations of one-dimensional probabilities

Chuang Xu; Arno Berger

doi:10.1016/j.jat.2019.03.005

Best finite constrained approximations of one-dimensional probabilities

Chuang Xu, Arno Berger^*

^*Corresponding author af dette arbejde

Institut for Matematiske Fag

2 Citationer (Scopus)

Abstract

This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the L ^r -Kantorovich (or transport) distance, where either the locations or the weights of the approximations’ atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) L ^r -functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach.

Originalsprog	Engelsk
Tidsskrift	Journal of Approximation Theory
Vol/bind	244
Sider (fra-til)	1-36
Antal sider	36
ISSN	0021-9045
DOI	https://doi.org/10.1016/j.jat.2019.03.005
Status	Udgivet - 2019

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10.1016/j.jat.2019.03.005

https://arxiv.org/pdf/1704.07871.pdfLicens: CC BY-NC-ND

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title = "Best finite constrained approximations of one-dimensional probabilities",

abstract = " This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the L r -Kantorovich (or transport) distance, where either the locations or the weights of the approximations{\textquoteright} atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) L r -functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach. ",

keywords = "Asymptotically best approximation, Balanced function, Best uniform approximation, Constrained approximation, Kantorovich distance, Quantile function",

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year = "2019",

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language = "English",

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journal = "Journal of Approximation Theory",

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AU - Xu, Chuang

AU - Berger, Arno

PY - 2019

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N2 - This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the L r -Kantorovich (or transport) distance, where either the locations or the weights of the approximations’ atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) L r -functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach.

AB - This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the L r -Kantorovich (or transport) distance, where either the locations or the weights of the approximations’ atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) L r -functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach.

KW - Asymptotically best approximation

KW - Balanced function

KW - Best uniform approximation

KW - Constrained approximation

KW - Kantorovich distance

KW - Quantile function

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DO - 10.1016/j.jat.2019.03.005

M3 - Journal article

AN - SCOPUS:85063510131

SN - 0021-9045

VL - 244

SP - 1

EP - 36

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

ER -

Best finite constrained approximations of one-dimensional probabilities

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