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

¹ Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland.
² Institute of Mathematics, Polish Academy of Sciences, ul. Kopernika 18, 51-617 Wrocław, Poland.
³ Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Cracov, Poland.

^** Corresponding author: cmielbog@gmail.com

Received: 9 February 2018
Accepted: 26 October 2018

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 Wyłupek [TEST 21 (2012) 730–756]. 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 definition 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.