Comparing two treatments in terms of the likelihood ratio order

TL;DR

This paper introduces new phi-divergence based test statistics for comparing two treatments under the likelihood ratio order, demonstrating their superior power in small to moderate samples through theoretical and simulation analysis.

Contribution

It develops a new family of test statistics extending classical tests, with proven asymptotic distribution and improved performance in specific scenarios.

Findings

New test statistics outperform classical tests in power for small/moderate samples.

Asymptotic distribution is a chi-bar random variable.

Wilcoxon test is less effective than likelihood ratio and Pearson tests.

Abstract

In this paper new families of test statistics are introduced and studied for the problem of comparing two treatments in terms of the likelihood ratio order. The considered families are based on phi-divergence measures and arise as natural extensions of the classical likelihood ratio test and Pearson test statistics. It is proven that their asymptotic distribution is a common chi-bar random variable. An illustrative example is presented and the performance of these statistics is analysed through a simulation study. Through a simulation study it is shown that, for most of the proposed scenarios adjusted to be small or moderate, some members of this new family of test-statistic display clearly better performance with respect to the power in comparison to the classical likelihood ratio and the Pearson's chi-square test while the exact size remains closed to the nominal size. In view of the exact powers and significance levels, the study also shows that the Wilcoxon test-statistic is not as good as the two classical test-statistics.