Improved estimator of the entropy and goodness of fit tests in ranked set sampling

TL;DR

This paper introduces improved entropy estimators using Ranked Set Sampling (RSS), demonstrating their superiority over simple random sampling in bias, RMSE, and power for goodness of fit tests for exponential and normal distributions.

Contribution

The paper develops two new entropy estimators within RSS, compares them with existing estimators, and applies them to enhance goodness of fit tests, showing increased power.

Findings

RSS estimators outperform SRS in bias and RMSE.

RSS-based tests have higher power for exponential and normal distributions.

The best estimator improves entropy and KL information estimation.

Abstract

The entropy is one of the most applicable uncertainty measures in many statistical and en- gineering problems. In statistical literature, the entropy is used in calculation of the Kullback- Leibler (KL) information which is a powerful mean for performing goodness of fit tests. Ranked Set Sampling (RSS) seems to provide improved estimators of many parameters of the popu- lation in the huge studied problems in the literature. It is developed for situations where the variable of interest is difficult or expensive to measure, but where ranking in small sub-samples is easy. In This paper, we introduced two estimators for the entropy and compare them with each other and the estimator of the entropy in Simple Random Sampling (SRS) in the sense of bias and Root of Mean Square Errors (RMSE). It is observed that the RSS scheme would improve this estimator. The best estimator of the entropy is used along with the estimator of the mean and two biased and unbiased estimators of variance based on RSS scheme, to esti- mate the KL information and perform goodness of fit tests for exponentiality and normality. The desired critical values and powers are calculated. It is also observed that RSS estimators would increase powers.