On Aly's Permutation Test for Variance Equality - Wojciech GAMROT (Communications of International Proceedings, 2021)
There are numerous significance tests described in the statistical literature, that may be used to compare variances of probability distributions. One of them is Aly's permutation test. It verifies the variance equality hypothesis by confronting the test statistic with an estimate of its distribution that is obtained through repeatedly exchanging data between two independent samples and re‑calculating the statistic. This leads to the assessment of the achieved significance level and to an eventual rejection of the hypothesis. In this paper a pair of distributions for which Aly's test performs very poorly is identified. By conveniently choosing parameters of these distributions one may make the probability of rejecting a true null hypothesis arbitrarily close to one.
Keywords: Permutation Test; Aly's Test; Variance Comparison; Sample Replication