Sample Estimate expression in Brunner-Muenzel test with R (lawstat)

#1
Dear Talk Stats Community,

I try to provide a reasonable interpretation of Sample Estimate expression P(X<Y) + .5*P(X=Y), which is shown as an output for BM-test in R (lawstat package). I am confused with the relationship between the following three probabilities: P(X<Y), P(X>Y), P(X=Y). Initially, I have assumed that X<Y and X=Y are mutually exclusive situations, so that P(X<Y) = 1 - P(X=Y). Now I doubt the propriety of this relationship and consider a possibility that P(X<Y) + P(X>Y) + P(X=Y) = 1. Neither of these scenarios, however, accounts for Sample Estimate = 0.5 in the absence of a true difference. It also contradicts with the way H0 is formulated in BM test (H0: P(X<Y)=0.5; H1: P(X<Y)>0.5, for our attached example). If there is no difference, it seems reasonable to assume that P(X=Y) is equal to 1, while P(X<Y) is zero. Our hypothesis tells us that P(X<Y) = P(X>Y) = 0.5 in the absence of a true difference.

Would you help me understand where is the fault in my reasoning?

Thank you.

Igor.
 

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