Asymptotic Normality of Simple Linear Rank Statistics Under Alternatives II

Abstract

This is a straightforward continuation of Hajek (1968). We provide a further extension of the Chernoff-Savage (1958) limit theorem. The requirements concerning the scores-generating function are relaxed to a minimum: we assume that this function is a difference of two non-decreasing and square integrable functions. Thus, in contradistinction to Hajek (1968), we dropped the assumption of absolute continuity. The main results are accumulated in Section 2 without proofs. The proofs are given in Sections 4 through 7. Section 3 contains auxiliary results.