Order Statistics Estimators of the Location of the Cauchy Distribution
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基于顺序统计量的柯西分布位置估计
- 研究问题
- 在不超过二十个观测的小样本中，柯西分布位置参数的不同估计量在操作便利性和有限样本效率上如何比较？
- 方法与思路
- 作者主要考察基于顺序统计量的一般线性估计量，以 Cramér–Rao 下界与估计方差之比衡量小样本效率。
- 材料陈述的贡献
- 摘要称，文章比较多种估计量的小样本实用性，并在需要时汇总已有最大似然结果；具体比较结论未给出。
- 阅读关注点
- 应区分摘要引述的既有渐近结论与本文的小样本贡献，并核对权重构造、方差存在性及效率计算范围。
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- 仅据版本未注明的 OpenAlex 摘要；效率公式含疑似乱码，需查阅原 PDF；不能将引述的既有结果视为本文新发现。
原文摘要
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].