Bounds on asymptotic relative efficiencies of robust estimates of locations for contaminations by scale mixtures
type: Item
instance of: scholarly article
title: Bounds on asymptotic relative efficiencies of robust estimates of locations for contaminations by scale mixtures
zbMATH Open document ID: 0804.62036
DOI: 10.1016/0378-3758(94)90155-4
author: Anirban DasGupta
published in: Journal of Statistical Planning and Inference
publication date: 1995-01-19T00:00:00Z
review text: The paper deals with robust estimation of a single location parameter \(\theta\) from a sample \(X_ 1, X_ 2,\) \(\dots, X_ n\) with \(X_ i= \theta+ Z_ i\). The errors \(Z_ i\) are independent and identically distributed, each with a cumulative distribution function \(F\) belonging to the generalized Tukey model: \[ {\mathcal F}= \bigl\{F:\;F(x)=\int [(1- \varepsilon_ 0) H(x)+ \varepsilon_ 0 H(x/ \sqrt{s})] dG(s)\bigr\}; \] \(G[s_ 1,\infty)=1\), \(\int s dG(s)= s_ 0\), with \(\varepsilon_ 0\), \(s_ 0\), \(s_ 1\) fixed, \(H\) being symmetric and absolutely continuous. The objective is to find the infimum and the supremum of the asymptotic relative efficiency of three well-known robust estimators: the Hodges - Lehmann, the sample median and the trimmed mean, with respect to the sample mean. It is interesting that these infimums and supremums remain bounded and in fact, very often are quite close to each other. As a consequence, the author is also in a position to answer the general question: given \(s_ 0\), how large is a contamination \(\varepsilon_ 0\) required for the infimum of the efficiency over \({\mathcal F}\) to exceed 1.
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62G35
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 638282
zbMATH Keywords: Hodges-Lehmann estimator
zbMATH Keywords: robust estimation
zbMATH Keywords: location
zbMATH Keywords: cumulative distribution
zbMATH Keywords: generalized Tukey model
zbMATH Keywords: infimum
zbMATH Keywords: supremum
zbMATH Keywords: asymptotic relative efficiency
zbMATH Keywords: sample median
zbMATH Keywords: trimmed mean
zbMATH Keywords: sample mean
reviewed by: Spiridon Penev
MaRDI profile type: Publication
full work available at URL: 0378-3758(94)90155-4
OpenAlex ID: W2003586941
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description: scientific article; zbMATH DE number 638282