I have been given a homework in a subject called "Non-Parametric Statistics" and I'm a bit stuck with it. I would be very thankful if you could give me any advice or help, which would lead to a solution!
The task is as follows:
Show that if $F_x$ is symmetrical and the relation between distribution functions $F_x(x)$ and $G_y(y)$ is $$ F_x(x) = G_y(x - \Delta)$$, where $\Delta$ > 0, then
$$ P(X > Y, X' > Y) = P(X > Y, X > Y') $$, whereas $X'$ is from the same distribution as $X$ and $Y'$ is from the same distribution as $Y$. A hint that has been given: Without loss of generality we may assume, that $F_x$ is symmetrical with respect to 0. In this case $X$ and $-X$ are from the same distribution.
We have this homework given in the Wilcoxon Rank Sum Test part, but in general case the solution idea stays the same.
My first thought was to use the independence of variables, so I could give $P(X > Y, X' > Y)$ like this $P(X > Y)$*$P(X' > Y)$ as a product, but I wasn't sure about it, whether they are independent or not ..
Thanks in advance!