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

Abstract

Three self-normalized two-sample test statistics are proposed for testing whether or not ${\mu }_i/{\sigma }_i^v,$ i = 1, 2, are equal for a given v > 0, where μ_i and ${\sigma }_i^2$ 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.

Keywords:

• Asymptotic normality
• coefficient of variation
• college students’ grades
• distribution free
• height difference
• variance-to-mean ratio
• weight difference

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 62F03
• 62G20

Acknowledgments

The authors would like to thank the two anonymous referees for their helpful comments and suggestions, which have led to the improvement of this paper.

Additional information

Funding

The research was supported by the National Natural Science Foundation of China (General Program, No. 71873128 and 11571337; Young Scholars Program, No. 71201152; Key Program, No. 71631006) and the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN-2017-05720).