Sankhya: The Indian Journal of Statistics

1996, Volume 58, Series B, Pt. 1, pp. 38--44



KHURSHEED ALAM, Clemson University
AMITAVA MITRA, Auburn University

SUMMARY.  Robust methods of estimation have been developed to modify the least squares estimates so that outliers have reduced influence on the estimates. The outliers may arise from gross errors or from distributions with long tails. Statistics which may be represented as a linear combination of order statistics, called $L$-estimates, provide a class of robust estimates. The trimmed mean is called adaptive when the trimming proportion is determined from the data. The trimming proportion may be determined by the tail-length of the underlying distribution. Hogg (1967) considered the kurtosis as a measure of the tail-length, and used the sample kurtosis to determine the trimming proportions of a combination of trimmed means for estimating the center of a symmetric distribution. In a subsequent paper on adaptive robust estimation, Hogg (1974) considered the choice of the trimming proportion, based on a ratio of two $L$-estimates, as a measure of the tail-length of a distribution. In this paper we review Hogg's method and present certain empirical results on the efficiency and robustness of the given method. In this context we derive a result of related interest on the asymptotic property of the joint distribution of two $L$-estimates.

AMS (1980) subject classification. 62F35.

Key words and phrases. Adaptive robust estimation, $L$-estimate, tail-length, peakedness, location parameter.

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