Comparing ratios of the mean to a power of variance in two samples via self-normalized test statistics

Three self-normalized two-sample test statistics are proposed for testing whether or not μi/σiv, i = 1, 2, are equal for a given v > 0, where μi and σi2 are the common mean and variance of the ith sample for i = 1, 2. They can be applied, but not limited to, the problems of the testing the equality of coefficients of variance when v = 1 and the equality of variance-to-mean ratio when v = 2. The self-normalized two-sample test statistics are distribution free and only dependent on the sample means and sample variances of both samples. It is shown that these test statistics are asymptotically normally distributed under the null hypothesis for each fixed v > 0. Simulation studies support the theoretical results. In addition, we illustrate the applicability of the proposed test by comparing students’ grades in seven semesters at a university in China and comparing the heights as well as weights of males to those of females, who have had their annual health examination at a health center in China.