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January, 1976 Robust Estimation of a Location Parameter in the Presence of Asymmetry
John R. Collins
Ann. Statist. 4(1): 68-85 (January, 1976). DOI: 10.1214/aos/1176343348

Abstract

Huber's theory of robust estimation of a location parameter is adapted to obtain estimators that are robust against a class of asymmetric departures from normality. Let F be a distribution function that is governed by the standard normal density on the set [d,d] and is otherwise arbitrary. Let X1,,Xn be a random sample from F(xθ), where θ is the unknown location parameter. If ψ is in a class of continuous skew-symmetric functions Ψc which vanish outside a certain set [c,c], then the estimator Tn, obtained by solving ψ(XiTn)=0 by Newton's method with the sample median as starting value, is a consistent estimator of θ. Also n12(Tnθ) is asymptotically normal. For a model of symmetric contamination of the normal center of F, an asymptotic minimax variance problem is solved for the optimal ψ. The solution has the form ψ(x)=x for |x|x0,ψ(x)=x1tanh[12x1(c|x|)]sgn(x) for x0|x|c, and ψ(x)=0 for |x|c. The results are extended to include an unknown scale parameter in the model.

Citation

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John R. Collins. "Robust Estimation of a Location Parameter in the Presence of Asymmetry." Ann. Statist. 4 (1) 68 - 85, January, 1976. https://doi.org/10.1214/aos/1176343348

Information

Published: January, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0351.62035
MathSciNet: MR400538
Digital Object Identifier: 10.1214/aos/1176343348

Subjects:
Primary: 62G35
Secondary: 62G05 , 62G20

Keywords: M-estimators , asymmetry , location parameter , minimax variance , robust estimation

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 1 • January, 1976
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