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 ( F Y , F X ) of the dominance of Y over X. Thus we label the central point at which the graph of F X ( x ) meets the graph of 1 - F Y ( x ) as ( x dom ( F Y , F X ) , dom ( F Y , F X ) ) . Conditions are given for the asymptotic normality of ( x dom ( 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.