Centering of Signed Rank Statistics with a Continuous Score-Generating Function

Abstract

For a continuous score generating function, Hájek [2] established the asymptotic normality of a simple linear rank statistic $S_N $ with natural parameters $({\bf E}S_N ,{\operatorname{Var}}S_N )$ as well as $({\bf E}S_N ,\sigma _N^2 )$, where $\sigma _N^2 $ is some constant. The permissibility of replacing ${\bf E}S_N $ by a simpler constant $\mu _N $ was shown by Hoeffding [4] under conditions slightly stronger than Hájek’s. Following Hájek’s methods, Hušková [5] derived the asymptotic normality of a simple signed rank statistic $S_N^ + $ with parameters $({\bf E}S_N^ + ,{\operatorname{Var}}S_N^ + )$ as well as $({\bf E}S_N^2 ,\sigma _N^2 )$ and left open the problem of the replacement of ${\bf E}S_N^ + $ by some simpler constant. In this note we close this problem of the replacement of ${\bf E}S_N^ + $ by a simpler constant $\mu _N^ + $. The solution is a follow-up of Hoeffding [4]. We also provide a slight generalization with regard to the choice of scores.