A Bootstrap Test for Comparing Two Variances: Simulation of Size and Power in Small Samples

摘要

An F statistic was proposed by Good and Chernick (19937. Good , P. I. , Chernick , M. ( 1993 ). Testing the equality of variances of two populations. Unpublished manuscript .View all references) in an unpublished paper, to test the hypothesis of the equality of variances from two independent groups using the bootstrap; see Hall and Padmanabhan (19978. Hall , P. , Padmanabhan , A. R. ( 1997 ). Adaptive inference for the two sample problem . Technometrics 39 : 412 - 422 .[Taylor & Francis Online], [Web of Science ®]View all references), for a published reference where Good and Chernick (19937. Good , P. I. , Chernick , M. ( 1993 ). Testing the equality of variances of two populations. Unpublished manuscript .View all references) is discussed.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 (20101. Chernick , M. R. , LaBudde , R. A. ( 2010 ). Revisiting qualms about bootstrap confidence intervals . American Journal of Mathematical and Management Sciences 29 : 437 - 456 .View all references) and Schenker (198513. Schenker , N. ( 1985 ). Qualms about bootstrap confidence intervals . Journal of the American Statistical Association 80 : 360 - 361 .[Taylor & Francis Online], [Web of Science ®]View all references) 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 (20101. Chernick , M. R. , LaBudde , R. A. ( 2010 ). Revisiting qualms about bootstrap confidence intervals . American Journal of Mathematical and Management Sciences 29 : 437 - 456 .View all references). 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.View full textDownload full textKey WordsBCa, Efron's percentile method, Normal-t , Schenker's F Related var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; Add to shortlist Link Permalink http://dx.doi.org/10.1080/10543406.2011.611082