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A Note on the Estimation of the Location Parameter of the Cauchy Distribution

Daniel Bloch
Journal of the American Statistical Association
Vol. 61, No. 315 (Sep., 1966), pp. 852-855
DOI: 10.2307/2282794
Stable URL: http://www.jstor.org/stable/2282794
Page Count: 4
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A Note on the Estimation of the Location Parameter of the Cauchy Distribution
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Abstract

Recently Professors T. J. Rothenberg, F. M. Fisher, and C. B. Tilanus published a paper proposing the class of trimmed means as estimators of the location parameter of the Cauchy distribution [5]. They showed that the asymptotic sampling variance of the estimators in this class is essentially minimized by using the middle 24% of the sample order statistics. The corresponding estimate has an asymptotic relative efficiency to the best estimator for complete samples (A.R.E.) of .87796 as compared to an A.R.E. of .81057 for the sample median. In this paper a few "quick estimators" are considered as estimators for the location parameter of the Cauchy Distribution. A "quick estimate" is a linear combination (a weighted average) of one or more order statistics. Our goal is to find a simple estimator, i.e. an estimator based on only a few order statistics, which has an A.R.E. of at least 90%. We found an estimator based on five order statistics which is considerably better than the optimum trimmed mean (using the middle 24% of the sample order statistics) and much better than the sample median. The A.R.E. of the optimum censored estimate with censored fractiles .38 and .62 is also found, and a comparison between the trimmed, censored, and proposed estimators is made.

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