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Robust M-estimators of location vectors

Author

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  • Collins, John R.

Abstract

Let X1,...,Xn be i.i.d. random vectors in Rm, let [theta][epsilon]Rm be an unknown location parameter, and assume that the restriction of the distribution of X1-[theta] to a sphere of radius d belongs to a specified neighborhood 1 of distributions spherically symmetric about 0. Under regularity conditions on 1 and d, the parameter [theta] in this model is identifiable, and consistent M-estimators of [theta] (i.e., solutions of [Sigma]i=1n[psi](Xi-[theta])(Xi-[theta])=0) are obtained by using "re-descenders," i.e., [psi]'s wh satisfy [psi](x)=0 for x>=c. An iterative method for solving for is shown to produce consistent and asymptotically normal estimates of [theta] under all distributions in 1. The following asymptotic robustness problem is considered: finding the [psi] which is best among the re-descenders according to Huber's minimax variance criterion.

Suggested Citation

  • Collins, John R., 1982. "Robust M-estimators of location vectors," Journal of Multivariate Analysis, Elsevier, vol. 12(4), pages 480-492, December.
    • Handle: RePEc:eee:jmvana:v:12:y:1982:i:4:p:480-492
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