Robust M- and L-Estimators of Scale Parameter

Abstract

We consider first the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at the error distribution of the shrinking contamination neighborhood and derive an expression for the maximal asymptotic mean-squared-error, for a suitably regular score function, followed by a lower bound on it. We next show that the minimax asymptotic mean-squzred-error is attained at an M-estimator of scale with the truncated MLE score function which, when specialized to the Standard Normal error distribution has the form of Huber's Proposal 2. The latter minimax property is also shown to hold for α-trimmed variance as an L-estimator of scale.

Keywords:

• Robust estimation
• Scale parameter
• M-estimators
• L-estimators
• α-trimmed variance
• Minimax asymptotic
• Mean-squared-error

Mathematics Subject Classification:

• 62F35
• 62F12

Acknowledgment

The author would like to thank Dr. J.R. Collins of the University of Calgary for his many helpful discussions with me about this topic.