Efficient and adaptive nonparametric test for the two-sample problem

The notion of efficient test for a Euclidean parameter in a semiparametric model was introduced by Stein [Proc. Third Berkeley Symp. Math. Statist. Probab. 1 (1956) 187-195]. Such tests are locally most powerful for a wide class of infinite-dimensional nuisance parameters. The first formal application of this notion to a suitably parametrized twosample problem was provided by Hajek [Ann. Math. Statist. 33 (1962) 1124-1147]. However, this and subsequent solutions appear to be not well-suited for practical applications. This article aims to show that an adaptive two-sample test introduced recently by Janic-Wr?blewska and Ledwina [Scand. J. Statist. 27 (2000) 281-297] is locally most powerful under a more realistic setting.