Combining Stochastic Tendency and Distribution Overlap Towards Improved Nonparametric Effect Measures and Inference

TL;DR

This paper enhances nonparametric effect measures by combining the Mann-Whitney functional with a distribution overlap index, leading to improved inference, larger consistency regions, and more interpretable effect measures.

Contribution

It introduces a novel joint functional combining ${ heta}$ and $I_2$, extending the Wilcoxon-Mann-Whitney test with better power and interpretability.

Findings

The joint asymptotic distribution of estimators is derived.

The new test has larger consistency regions and maintains competitive power.

Effect measures are more straightforward to interpret.

Abstract

A fundamental functional in nonparametric statistics is the Mann-Whitney functional ${\theta} = P (X < Y )$ , which constitutes the basis for the most popular nonparametric procedures. The functional ${\theta}$ measures a location or stochastic tendency effect between two distributions. A limitation of ${\theta}$ is its inability to capture scale differences. If differences of this nature are to be detected, specific tests for scale or omnibus tests need to be employed. However, the latter often suffer from low power, and they do not yield interpretable effect measures. In this manuscript, we extend ${\theta}$ by additionally incorporating the recently introduced distribution overlap index (nonparametric dispersion measure) $I_2$ that can be expressed in terms of the quantile process. We derive the joint asymptotic distribution of the respective estimators of ${\theta}$ and $I_2$ and construct confidence regions. Extending the Wilcoxon- Mann-Whitney test, we introduce a new test based on the joint use of these functionals. It results in much larger consistency regions while maintaining competitive power to the rank sum test for situations in which {\theta} alone would suffice. Compared with classical omnibus tests, the simulated power is much improved. Additionally, the newly proposed inference method yields effect measures whose interpretation is surprisingly straightforward.