Some Pitman Closeness Properties Pertinent to Symmetric Populations

TL;DR

This paper investigates Pitman closeness probabilities for symmetric estimators, establishing conditions for estimator comparison, properties of the sample median, and optimal ranked set sampling schemes for estimating the population median.

Contribution

It provides necessary and sufficient conditions for Pitman closeness comparison, and identifies the median and randomized median RSS as optimal estimators under symmetry.

Findings

Sample median is Pitman closer to the population median than other symmetric estimators.

The median and randomized median RSS are optimal sampling schemes for symmetric populations.

Conditions for estimator Pitman closeness are explicitly characterized.

Abstract

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter $\theta$. We first consider two symmetric estimators $\hat{\theta}_1$ and $\hat{\theta}_2$ and obtain necessary and sufficient conditions for $\hat{\theta}_1$ to be Pitman closer to the common median $\theta$ than $\hat{\theta}_2$. We then establish some properties in the context of estimation under Pitman closeness criterion. We define a Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to $\theta$ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other symmetrically distributed estimator of $\theta$. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.