For testing variability in the two-sample case, one may choose some pairs of observations in each sample and calculate the absolute difference for each pair, and then apply the Wilcoxon-Mann-Whitney test using the absolute differences from the two samples. Blair and Thompson (1992) considered this approach and proposed to choose the pairs randomly while keeping the pairs disjoint. Blair and Thompson's procedure was later criticized in Ramsey and Ramsey (2007) for the random selection and using only half of the data. If one makes use of all possible pairs in Blair and Thompson's Wilcoxon-Mann-Whitney procedure, then the problem of random selection and using only half of the data can be eliminated. However, the test statistic is no longer an ancillary under the null hypothesis that the two samples are from the same distribution. I propose to consider the Wilcoxon-Mann-Whitney test statistic based on all possible absolute differences from the two samples, search for the distribution that gives the most conservative critical value, and use the value as the critical value. The distribution that gives the most conservative critical value is to be found numerically using B-spline approximation for the density function of the sample distribution.
Keywords: Test for variability; Two-sample; Wilcoxon-Mann-Whitney
Biography: Dr. Huang is an associate professor of the department of statistics at National Chengchi University in Taiwan.