On a simple measure of dominance

One way to describe how much better treatment Y is than treatment X is to find a response value such that the probability of Y being better than this value is the same as the probability of X being worse. We consider the size of this probability as a measure dom(FY,FX) of the dominance of Y over X. Thus we label the central point at which the graph of FX(x) meets the graph of 1-FY(x) as (xdom(FY,FX),dom(FY,FX)). Conditions are given for the asymptotic normality of (xdom(G^,F^),dom(G^,F^)) when G^ and F^ are possibly dependent empirical distributions. A Wilson-type approximate confidence interval for dom(G,F) is proposed and this interval is shown to perform generally better than a Wald-type interval.