A formula for P(bi≤Ri≤ai, 1≤i≤m) under a general class of alternatives

Let Xi≤⋯≤Xm and Yi≤⋯≤Yn be two sets of independent order statistics from continous distributions with distribution functions F and G respectively. Let Ri denote the rank of Xi in the combined order sample. Steck (1980) has found an expression for P(bi≤Ri≤ai, all i) when F = h(G), h being the incomplete beta function with parameters (α,β−α+1). An alternative expression for the same probability is obtained which is computationally a substantial improvement on Steck's result.