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March, 1984 Finite Sample Breakdown of $M$- and $P$-Estimators
Peter J. Huber
Ann. Statist. 12(1): 119-126 (March, 1984). DOI: 10.1214/aos/1176346396

Abstract

The finite sample breakdown properties of $M$-estimators, defined by $\sum\rho(x_i - T) = \min!$, and of the associated Pitman-type or $P$-estimators, defined by $T = \frac{\int \exp\{-\Sigma \rho(x_i - \theta)\}\theta d\theta}{\int \exp\{-\Sigma\rho(x_i - \theta)\} d\theta},$ are investigated. If $\rho$ is symmetric, and $\psi = \rho'$ is monotone and bounded, then the breakdown point of either estimator is $\varepsilon^\ast = 1/2$. If $\psi$ decreases to 0 for large $x$ ("redescending estimators"), the same result remains true if $\rho$ is unbounded. For bounded $\rho$, the $P$-estimator is undefined, and the breakdown point of the $M$-estimator typically is slightly less than $1/2$; it is calculated in explicit form.

Citation

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Peter J. Huber. "Finite Sample Breakdown of $M$- and $P$-Estimators." Ann. Statist. 12 (1) 119 - 126, March, 1984. https://doi.org/10.1214/aos/1176346396

Information

Published: March, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0557.62034
MathSciNet: MR733503
Digital Object Identifier: 10.1214/aos/1176346396

Subjects:
Primary: 62F35

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 1 • March, 1984
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