Robust M-estimators of location vectors |
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Authors: | John R Collins |
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Institution: | University of Calgary, Calgary, Alberta, Canada |
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Abstract: | Let X1,…,Xn be i.i.d. random vectors in Rm, let θεRm be an unknown location parameter, and assume that the restriction of the distribution of X1−θ to a sphere of radius d belongs to a specified neighborhood
of distributions spherically symmetric about 0. Under regularity conditions on
and d, the parameter θ in this model is identifiable, and consistent M-estimators of θ (i.e., solutions of Σi=1nψ(|Xi−
|)(Xi−
)=0) are obtained by using “re-descenders,” i.e., ψ's wh satisfy ψ(x)=0 for x≥c. An iterative method for solving for
is shown to produce consistent and asymptotically normal estimates of θ under all distributions in
. The following asymptotic robustness problem is considered: finding the ψ which is best among the re-descenders according to Huber's minimax variance criterion. |
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Keywords: | Robust estimation location vector asymmetry asymptotically minimax |
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