A differential version of the Efron-Stein inequality: Bounding the variance of a function of an infinitely divisible variable
type: Item
instance of: scholarly article
title: A differential version of the Efron-Stein inequality: Bounding the variance of a function of an infinitely divisible variable
zbMATH Open document ID: 0664.60025
DOI: 10.1016/0167-7152(88)90034-X
author: Richard A. Vitale
published in: Statistics \& Probability Letters
publication date: 1988-01-01T00:00:00Z
review text: Let X be an infinitely divisible random variable with characteristic function in Lévy-Khinchine form: \[ \phi_ X(t)=\exp [itb-\sigma^ 2t^ 2/2+\int^{\infty}_{-\infty}(e^{itu}-1-itu/(1+u^ 2))(1+u^ 2)u^{-2} \nu (du)], \] where \(\nu\) is a finite measure with \(\nu (\{0\})=0\). If \(G(x)=\int^{x}_{x_ 0}g(u)dy+C_ 0\), then \[ var G(X)\leq \sigma^ 2E g^ 2(X)+\int^{\infty}_{-\infty}[u^{- 1}\int^{u}_{0}E g^ 2(X+y)dy](1+u^ 2)\nu (du). \] Equality occurs in the case of a finite upper bound if and only if \(G(X)=aX+b\) a.s. with some constants a and b. The result can be derived rom results of \textit{B. Efron} and \textit{C. Stein} [Ann. Stat. 9, 586-596 (1981; Zbl 0481.62035)].
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 4086641
zbMATH Keywords: infinitely divisible
zbMATH Keywords: characteristic function
MaRDI profile type: Publication
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full work available at URL: 0167-7152(88)90034-x
OpenAlex ID: W2091193732
description: scientific article; zbMATH DE number 4086641