Asymptotics of reweighted estimators of multivariate location and scatter. (English) Zbl 0957.62017

Lopuhaä, Hendrik P.

Ann. Stat. 27, No. 5, 1638-1665 (1999).

Summary: We investigate the asymptotic behavior of a weighted sample mean and covariance, where the weights are determined by the Mahalanobis distances with respect to initial robust estimators. We derive an explicit expansion for the weighted estimators. From this expansion it can be seen that reweighting does not improve the rate of convergence of the initial estimators. We also show that if one uses smooth \(S\)-estimators to determine the weights, the weighted estimators are asymptotically normal. Finally, we will compare the efficiency and local robustness of the reweighted \(S\)-estimators with two other improvements of \(S\)-estimators: \(\tau\)-estimators and constrained \(M\)-estimators.

MSC:

62F12	Asymptotic properties of parametric estimators
62H12	Estimation in multivariate analysis
62E20	Asymptotic distribution theory in statistics
62F35	Robustness and adaptive procedures (parametric inference)
62H10	Multivariate distribution of statistics

Keywords:

location; reweighted least squares; empirical processes; weighted sample mean; covariance; robust estimators

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