Exponentially Titled Empirical Distribution Function for Ranked Set Samples

TL;DR

This paper introduces exponentially tilted empirical likelihood estimators for the distribution function in ranked set sampling, providing new resampling algorithms and demonstrating their effectiveness through simulations and real data analysis.

Contribution

It proposes a novel ET estimator for the distribution function in ranked set samples and develops new resampling algorithms with proven asymptotic properties.

Findings

Bootstrap tests are asymptotically normal with small bias.

Algorithms perform well under perfect and imperfect ranking.

ET-based tests outperform empirical likelihood-based tests.

Abstract

We study nonparametric estimation of the distribution function (DF) of a continuous random variable based on a ranked set sampling design using the exponentially tilted (ET) empirical likelihood method. We propose ET estimators of the DF and use them to construct new resampling algorithms for unbalanced ranked set samples. We explore the properties of the proposed algorithms. For a hypothesis testing problem about the underlying population mean, we show that the bootstrap tests based on the ET estimators of the DF are asymptotically normal and exhibit a small bias of order $O(n^{-1})$. We illustrate the methods and evaluate the finite sample performance of the algorithms under both perfect and imperfect ranking schemes using a real data set and several Monte Carlo simulation studies. We compare the performance of the test statistics based on the ET estimators with those based on the empirical likelihood estimators.