Consider $X\sim N(1,1)$ and $Y\sim exp(1)$. The distributions are totally different, but they do have equal variance, so if I were to sample from these and test for unequal variance, I would not want to reject a null hypothesis of variance equality.
I am finding that the usual suspects for variance testing reject too often. In my Monte Carlo simulations, the F-test (
var.test in R), Brown-Forsythe (
car::leveneTest(…, center="median")), Levene (
car::leveneTest(…, center="mean")), Ansari-Bradley (
ansari.test), and permutation testing (my own custom function) all rejected considerably more often that my chosen $\alpha$. These tests are overpowered for this situation. I suspect (particularly for Ansari-Bradley) that this is due to the tests examining some kind of surrogate for variance (some kind of "scale" parameter) rather than variance itself, and they are finding that $X$ and $Y$ do differ on that surrogate.
What would be a test that has appropriate power for my $X$ and $Y?$