ROBUSTNESS COMPARISONS OF SOME CLASSES OF LOCATION PARAMETER ESTIMATORS

JOHN R. COLLINS

Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

(Received May 13, 1997; revised August 28, 1998)

Abstract. Asymptotic biases and variances of M-, L- and R-estimators of a location parameter are compared under ε-contamination of the known error distribution F_{0} by an unknown (and possibly asymmetric) distribution. For each ε-contamination neighborhood of F_{0}, the corresponding M-, L- and R-estimators which are asymptotically efficient at the least informative distribution are compared under asymmetric ε-contamination. Three scale-invariant versions of the M-estimator are studied:

(i) one using the interquartile range as a preliminary estimator of scale: (ii) another using the median absolute deviation as a preliminary estimator of scale; and (iii) simultaneous M-estimation of location and scale by Huber’s Proposal 2. A question considered for each case is: when are the maximal asymptotic biases and variances under asymmetric ε-contamination attained by unit point mass contamination at ∞? Numerical results for the case of the ε-contaminated normal distribution show that the L-estimators have generally better performance (for small to moderate values of ε) than all three of the scale-invariant M-estimators studied.

Key words and phrases: Robust estimation, M-, L- and R-estimators, asymptotic biases, asymptotic variances, asymmetric contamination.