On the Centering of a Simple Linear Rank Statistic

Hájek (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 μ, 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 ofES by μis permissible if one of Hájek’s conditions is slightly strengthened.