Abstract

A new class of statistics obtained by ordering the absolute values of the observations arising from absolutely continuous distributions which are symmetrically distributed about zero is introduced in this paper. The statistics generated by the above method are named as absolved order statistics (AOS) of the given sample. The association of the distribution of these statistics with the distribution of order statistics arising from the folded form of the parental density about zero is outlined. The vector of AOS is proved to be a minimal sufficient statistic for the class ${\mathcal{F}}_{\theta }^{(1)}$ of all absolutely continuous distributions which are symmetrically distributed about zero. A method of estimation of the scale parameter of any distribution belonging to ${\mathcal{F}}_{\theta }^{(1)}$ using AOS is described. Illustration on the advantage of the above method of estimation is described for the distributions such as (i) logistic, (ii) normal, and (iii) double Weibull. A more realistic censoring scheme involving AOS as well is discussed in this paper. We have derived the U-statistic estimator based on AOS for the scale parameter σ of any distribution $f(x,\sigma )\in {\mathcal{F}}_{\theta }^{(1)}$ using the best linear unbiased estimate (BLUE) based on AOS of a preliminary sample as kernel. We have illustrated the performance of this estimator with an U-statistic generated from BLUE based on order statistics for each of (i) logistic (ii) normal and (iii) double Weibull distributions.

摘要

本文介绍了一类新的统计量，该统计量是通过对绝对连续分布产生的观测值的绝对值进行排序而获得的，这些分布是关于零对称分布的。上述方法产生的统计量称为给定样本的无定序统计量（AOS）。概述了这些统计量的分布与由亲本密度大约为零的折叠形式产生的顺序统计量分布的关联。AOS 的向量被证明是该类的最小充分统计量${\mathcal{F}}_{\theta }^{(1)}$所有绝对连续分布的对称分布在零附近。一种估计属于任何分布的尺度参数的方法${\mathcal{F}}_{\theta }^{(1)}$描述了使用 AOS。描述了上述估计方法的优势，例如 (i) 逻辑分布、(ii) 正态分布和 (iii) 双 Weibull。本文还讨论了涉及 AOS 的更现实的审查方案。我们推导出了基于 AOS 的 U 统计量估计量，用于任何分布的尺度参数σ$F(X,\sigma )\in {\mathcal{F}}_{\theta }^{(1)}$使用基于初步样本的 AOS 的最佳线性无偏估计 (BLUE) 作为内核。我们已经使用基于 (i) 逻辑 (ii) 正态分布和 (iii) 双 Weibull 分布中的每一个的顺序统计量从 BLUE 生成的 U 统计量来说明该估计器的性能。