Post date: Nov 01, 2015 8:7:47 PM
It seems as though c-boxes for variances should be able to address this issue. But there are two wrinkles. First, no assumption is made about the shape or family of the distributions whose variances we're talking about. Second, the two variables are to be dependent, rather than independent. Presumably, the Fréchet operations might be appropriate for the second wrinkle, but I don't think we have a distribution-free estimator for variance. (Just as we also lack a distribution-free estimator for the mean. If we had distribution-free estimators for the mean, we would be able to solve the non-parametric UCL issue, which is another holy grail in environmental statistics.)
Rand Wilcox (2015). Comparing the variances of two dependent variables. Journal of Statistical Distributions and Applications 2: 7
Abstract: Various methods have been derived that are designed to test the hypothesis that two dependent variables have a common variance. Extant results indicate that all of these methods perform poorly in simulations. The paper provides a new perspective on why the Morgan-Pitman test does not control the probability of a Type I error when the marginal distributions have heavy tails. This new perspective suggests an alternative method for testing the hypothesis of equal variances and simulations indicate that it continues to perform well in situations where the Morgan-Pitman test performs poorly.
Keywords: Morgan-Pitman test; Heteroscedasticity; HC4 estimator; Well elderly 2 study