Intermediate efficiency in nonparametric testing problems with an application to some weighted statistics

TL;DR

This paper introduces a new variant of intermediate efficiency to evaluate nonparametric tests, demonstrating that a weighted stochastic dominance test often outperforms the classical Kolmogorov-Smirnov test in power, supported by theoretical and simulation results.

Contribution

It develops a general intermediate efficiency concept for nonparametric tests and applies it to compare a new weighted test with the classical Kolmogorov-Smirnov test, showing improved efficiency.

Findings

The new test is more efficient than the classical test for many alternatives.

The proposed efficiency measure accurately predicts test power.

Simulation results confirm the theoretical efficiency advantages.

Abstract

The basic motivation and primary goal of this paper is a qualitative evaluation of the performance of a new weighted statistic for a nonparametric test for stochastic dominance based on two samples, which was introduced in Ledwina and Wy{\l}upek (2012). For this purpose, we elaborate a useful variant of Kallenberg's notion of intermediate efficiency. This variant is general enough to be applicable to other nonparametric problems. We provide a formal defnition of the proposed variant of intermediate efficiency, describe the technical tools used in its calculation, and provide proofs of related asymptotic results. Next, we apply this approach to calculating the intermediate efficiency of the new test with respect to the classical one-sided Kolmogorov-Smirnov test, which is a recognized standard for this problem. It turns out that for a very large class of convergent alternatives the new test is more efficient than the classical one. We also report the results of an extensive simulation study on the powers of the tests considered, which shows that the new variant of intermediate efficiency reflects the exact behavior of the power well.