Monotonicity properties of the ordered ranks in the two-sample problem

Let X1,...,Xm and Y1,...,Yn be independent random samples from two absolutely continuous distributions F and G, respectively. For F=G, Fligner and Wolfe (1976) established some interesting properties of the Wi's, the number of X-observations between the (i-1)th and ith order statistics of the Y-sample. In particular, it follows from their results that when F=G, the Wi's are identically distributed. In this note we study this problem when the X's are greater than the Y's according to likelihood ratio and hazard rate orderings. It is shown that in both these cases, the Wi's exhibit stochastic increasing trends of different types.