The Annals of Statistics

Bias Robust Estimation of Scale

R. D. Martin and Ruben H. Zamar

Full-text: Open access

Abstract

In this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location-scale family. We define the class of $M$-estimates of scale with general location, and show that under certain regularity assumptions, these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution, and with respect to the $M$-estimate defining score function $\chi$. We establish expressions for the maximum asymptotic bias of $M$-estimates of scale over the contamination neighborhood as a function of the fraction of contamination. Using these expressions we construct asymptotically min-max bias robust estimates of scale. In particular, we show that a scaled version of the Madm (median of absolute residuals about the median) is approximately min-max bias-robust within the class of Huber's Proposal 2 joint estimates of location and scale. We also consider the larger class of $M$-estimates of scale with general location, and show that a scaled version of the Shorth (the shortest half of the data) is approximately min-max bias robust in this class. Finally, we present the results of a Monte Carlo study showing that the Shorth has attractive finite sample size mean squared error properties for contaminated Gaussian data.

Article information

Source
Ann. Statist., Volume 21, Number 2 (1993), 991-1017.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349161

Digital Object Identifier
doi:10.1214/aos/1176349161

Mathematical Reviews number (MathSciNet)
MR1232529

Zentralblatt MATH identifier
0787.62038

JSTOR
links.jstor.org

Subjects
Primary: 62F35: Robustness and adaptive procedures
Secondary: 62G35: Robustness

Keywords
Bias robustness $M$-estimates scale location Madm Shorth

Citation

Martin, R. D.; Zamar, Ruben H. Bias Robust Estimation of Scale. Ann. Statist. 21 (1993), no. 2, 991--1017. doi:10.1214/aos/1176349161. https://projecteuclid.org/euclid.aos/1176349161


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