A Generalized Stepwise Procedure with Improved Power for Multiple Inequalities Testing

Abstract

We propose a stepwise test, Step-SPA(⁠$k$⁠), for multiple inequalities testing. This test is analogous to the Step-SPA test of Hsu, Hsu, and Kuan (2010, J. Empirical Econ., 17, 471–484) but has asymptotic control of a generalized familywise error rate: the probability of at least $k$ false rejections. This test improves Step-RC(⁠$k$⁠) of Romano and Wolf (2007, Ann. Stat., 35, 1378–1408) by avoiding the least favorable configuration used in Step-RC(⁠$k$⁠). We show that the proposed Step-SPA(⁠$k$⁠) test is consistent and more powerful than Step-RC(⁠$k$⁠) under any power notion defined in Romano and Wolf (2005, Econometrica, 73, 1237–1282). An empirical study on Commodity Trading Advisor fund performance is then provided to illustrate the Step-SPA(⁠$k$⁠) test. Finally, we extend Step-SPA(⁠$k$⁠) to a procedure that asymptotically controls the false discovery proportion, the ratio of the number of false rejections over the number of total rejections, and show that it is more powerful than the corresponding procedure proposed by Romano and Wolf (2007, Ann. Stat., 35, 1378–1408).