I'm trying to compare two samples from multivariate normal distributions to see if their distributions are equivalent (within a factor of epsilon).
The standard version of this test is the energy test, but it is not useful for my purposes because it uses $P = Q$ as the null hypothesis, whereas I need $P \neq Q$ to be the null.
Some previous work has been done on this, mostly in this book, but nothing for my situation.
How can I extend the interval inclusion method described in this book to use the energy statistic? Should the test statistic used to get the confidence interval be the original test statistic (i.e. based on the null hypothesis $P = Q$), or do I have to derive my own based on the new null hypothesis? I think derive my own is definitely beyond my capabilities at this point.