An Adaptive Choice of the Scale Parameter for M-Estimators
STANFORD UNIV CA DEPT OF STATISTICS
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Let x sub 1 to x sub n be a random sample from a distribution symmetric about the unknown location parameter theta. A major class of robust estimators of location is the class of M-estimators, each of which corresponds to a function psi defined on the reals. To be scale equivariant, these estimators require the use of a scale equivariant function of the sample. Commonly, this scale parameter is chosen to be a constant times the sample MAD medial absolute deviation from the median. For a given function psi, the variance of the corresponding M-estimator vaires considerably with the value of the scale parameter. It is therefore proposed that the value which minimizes an estimate of the asymptotic variance of the M-estimator be used as the scaling factor. This adaptive method of scaling is shown to be asymptotically optimal under fairly general conditions, in the sense that the resulting M-estimator has the smallest possible asymptotic variance among all M-estimators based on psi. In particular, when the underlying distribution is normal, the adaptive estimator based on any reasonable psi achieves full asymptotic efficiency, i.e., is asymptotically equivalent to the sample mean. The performance of the estimator for small samples is investigated by Monte Carlo methods for several choices of psi using the triefficiency criteria. A slight modification of the above estimator compares favorably with Tukeys bisquare M-estimator for sample sizes as small as 20.
- Statistics and Probability