Necessary and sufficient conditions for weak consistency of the median of independent but not identically distributed random variables
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for weak consistency of the median of independent but not identically distributed random variables
zbMATH Open document ID: 0934.62052
author: Ivan Mizera
author: Jon A. Wellner
published in: The Annals of Statistics
publication date: 1999-11-09T00:00:00Z
review text: For each \(n=1,2,\dots\), suppose that \(X_{n1}\), \(X_{n2},\dots, X_{nn}\) are independent random variables with distribution functions \(F_{n1}\), \(F_{n2},\dots,F_{nn}\). Let \(\mathbb{F}_n\) denote the empirical distribution function of the \(X_{ni}\)'s: \[ \mathbb{F}_n(x)\equiv n^{-1}\sum^n_{i=1}1_{(-\infty, x]}(X_{ni}), \] and let \(\overline F_n\) be the average distribution function \[ \overline F_n\equiv n^{-1}\sum^n_{i=1}F_{ni}. \] For any distribution function \(G\), let \(G^{-1}\) be the left-continuous inverse of \(G\) defined by \(G^{-1}(u) \equiv\inf\{x:G(x)\geq u\}\), \(0<u<1\). Throughout this paper, unless otherwise noted, we call \(G^{-1}(1/2)\) the median of \(G\), even when there is a nondegenerate interval \([m_0,m_1]\) of median points in the sense that \(P_G(Y\leq m)\geq 1/2\) and \(P_G(Y\geq m)\geq 1/2\) for \(m\in [m_0,m_1]\). The problem in this paper is to give necessary and sufficient conditions for weak consistency of the sample median \(\mathbb{F}_n^{-1}(1/2)\): under what conditions on the \(F_{ni}\)'s does it hold that \(\mathbb{F}_n^{-1} (1/2)-\overline F_n^{-1}(1/2)\to_P0\)?
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62F12
zbMATH DE Number: 1359537
zbMATH Keywords: majorization
zbMATH Keywords: empirical distributions
zbMATH Keywords: weak consistency
zbMATH Keywords: median
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1028144854
OpenAlex ID: W1971191079
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DOI: 10.1214/AOS/1028144854
description: scientific article; zbMATH DE number 1359537