JSM 2005 Online Program

Abstract #303511

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Activity Number:	232
Type:	Contributed
Date/Time:	Tuesday, August 9, 2005 : 8:30 AM to 10:20 AM
Sponsor:	General Methodology
Abstract - #303511
Title:	Minimax Asymptotic Mean-squared-error of M-estimators of Scale
Author(s):	Daniela Szatmari-Voicu*+
Companies:	University of Calgary
Address:	123 Copperfield Lane SE, Calgary, AB, T2Z 4S9, Canada
Keywords:	robustness ; scale parameter ; minimax asymptotic mean-squared-error ; "shrinked" contamination neighborhood
Abstract:	Consider the scale estimation problem under the location-scale model X1, X2,..., Xn ~ Fn ((x-?)/s)), where -8 < ? < 8 and s > 0 are unknown and Fn is an unknown member of the "shrinked" neighborhood Fen = {Fn \| Fn = (1 - en}Fo + en G, where en = k/Vn, k = cst.> 0, n the sample size, Fo known error distribution and G unknown distribution} and the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at Fo. On this class of estimators, we derive an expression for the AMSE (maximal asymptotic mean-squared-error), for a suitably regular score function ? and error distribution Fo, followed by a lower bound on it. We next show the minimum AMSE is attained at an M-estimator of scale with the truncated MLE-score function and the minimum AMSE for Fo = F (Normal) has the form of Huber's Proposal 2. Calculations of AMSE for Huber's Proposal 2 are presented both in the case of Fo -Normal (where Huber's Proposal 2 is minimax asymptotic mean-squared-error) and Fo nonNormal. The minimum AMSE property for Huber's Proposal 2 for Fo = F is also shown to hold for a particular L-estimator of scale, namely the a-trimmed variance.

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