A bootstrap test for comparing two variances: simulation of size and power in small samples

An F statistic was proposed by Good and Chernick ( 1993 ) in an unpublished paper, to test the hypothesis of the equality of variances from two independent groups using the bootstrap; see Hall and Padmanabhan ( 1997 ), for a published reference where Good and Chernick ( 1993 ) is discussed. We look at various forms of bootstrap tests that use the F statistic to see whether any or all of them maintain the nominal size of the test over a variety of population distributions when the sample size is small. Chernick and LaBudde ( 2010 ) and Schenker ( 1985 ) showed that bootstrap confidence intervals for variances tend to provide considerably less coverage than their theoretical asymptotic coverage for skewed population distributions such as a chi-squared with 10 degrees of freedom or less or a log-normal distribution. The same difficulties may be also be expected when looking at the ratio of two variances. Since bootstrap tests are related to constructing confidence intervals for the ratio of variances, we simulated the performance of these tests when the population distributions are gamma(2,3), uniform(0,1), Student's t distribution with 10 degrees of freedom (df), normal(0,1), and log-normal(0,1) similar to those used in Chernick and LaBudde ( 2010 ). We find, surprisingly, that the results for the size of the tests are valid (reasonably close to the asymptotic value) for all the various bootstrap tests. Hence we also conducted a power comparison, and we find that bootstrap tests appear to have reasonable power for testing equivalence of variances.