On uncertainty and information properties of ranked set samples

TL;DR

This paper investigates the information properties of ranked set sampling under perfect and imperfect ranking, demonstrating its superiority over simple random sampling in terms of Shannon, Rényi, and KL information measures, and analyzing the impact of ranking errors.

Contribution

It provides a comprehensive analysis of the uncertainty and information content of ranked set samples using various entropy measures, including the effects of ranking errors.

Findings

Ranked set sampling has higher Fisher information than simple random sampling.

Information content increases with set size in ranked set sampling.

Ranking errors reduce the information content of the samples.

Abstract

Ranked set sampling is a sampling design which has a wide range of applications in industrial statistics, and environmental and ecological studies, etc.. It is well known that ranked set samples provide more Fisher information than simple random samples of the same size about the unknown parameters of the underlying distribution in parametric inferences. In this paper, we consider the uncertainty and information content of ranked set samples in both perfect and imperfect ranking scenarios in terms of Shannon entropy, R\'enyi and Kullback-Leibler (KL) information measures. It is proved that under these information measures, ranked set sampling design performs better than its simple random sampling counterpart of the same size. The information content is also a monotone function of the set size in ranked set sampling. Moreover, the effect of ranking error on the information content of the data is investigated.