Order Statistics Estimators of the Location of the Cauchy Distribution

Abstract

Abstract In a recent paper in this Journal, Rothenberg, Fisher and Tilanus [1] discuss a class of estimators of the location parameter of the Cauchy distribution, taking the form of the arithmetic average of a central subset of the sample order statistics. They show that the average of roughly the middle quarter of the ordered sample has minimum asymptotic variance within this class, and that asymptotically it eliminates about 36 per cent of the efficiency loss of the median (the most commonly used estimator) in comparison to the maximum likelihood estimator (m.l.e.). Of course both the m.l.e. and the best linear unbiased estimator based on the order statistics (BLUE) achieve full asymptotic efficiency in the Cramér-Rao sense and there can be no dispute about the relative merits of the three estimators asymptotically, or about the inferiority of the median (with asymptotic efficiency 8/π2 ė 0.8 compared with about 0.88 for the estimator of Rothenberg et al.). In any practical situation however, we will be concerned with estimation from samples of finite size and asymptotic properties will not necessarily give any guidance here. We are essentially concerned with two points in assessing the relative merits of estimators in small samples, their ease of application and "small-sample efficiency" which is conveniently measured as the ratio of the Cramér-Rao lower bound to the variance of the estimator. In this paper various estimators of the location of the Cauchy distribution are compared in these two respects for samples of up to 20 observations. The small-sample properties of the m.l.e. have been extensively discussed elsewhere (Barnett [2]) and relevant results are summarized where necessary. The main purpose of the paper is to discuss general linear estimators based on the order statistics, and to assess their utility in the present context. Since this paper was prepared a further interesting 'quick estimator', based on order statistics, for the location of the Cauchy distribution has been suggested by Bloch [8].

Keywords

Affiliated Institutions

• University of Alabama at Birmingham US

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