On the Centering of a Simple Linear Rank Statistic

Abstract

Hajek (1968) proved that under weak conditions the distribution of a simple linear rank statistic $S$ is asymptotically normal, centered at the mean $ES$. He left open the question whether under the same conditions the centering constant $ES$ may be replaced by a simpler constant $\mu$, as was found to be true in the two-sample case and under different conditions by Chernoff and Savage (1958) and Govindarajulu, LeCam and Rhagavachari (1966). In this paper it is shown that the replacement of $ES$ by $\mu$ is permissible if one of Hajek's conditions is slightly strengthened.