Abstract
This article considers the problem of testing the equality of the cumulative probabilities of two independent normal distributions. This is equivalent to the problem of testing the equality of the coefficients of variations from two separate populations, which in turn is equivalent to the problem of testing the equality of two Sharpe ratios in financial analyses when it is assumed that the data are normally distributed. This is a problem that has received considerable attention in the statistical and financial research literature; although, while previous approaches have relied upon large sample asymptotic results, the objective of this article is to develop a test procedure that guarantees the nominal confidence level for small sample sizes. Examples are provided to demonstrate the new test procedure, and software is available for its implementation from the authors.
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References
Amiri, S., and S. Zwanzig. 2010. An improvement of the nonparametric boot-strap test for the comparison of the coefficient of variations. Commun. Stat. Simulation Comput., 39(9), 1726–1734.
Bai, Z., K. Wang, and W. K. Wong. 2011. The mean-variance ratio test—A complement to the coefficient of variation test and the Sharpe ratio test. Stat. Prob. Lett., 81, 1078–1085.
Banik, S., and B. M. G. Kibria. 2011. Estimating the population coefficient of variation by confidence intervals. Commun. Stat. Simulation Comput., 40(8), 1236–1261.
Bennett, B. M. 1976. On an approximate test for homogeneity of coefficients of variation. In Ziegler, W. J. (ed.), Contributions to Applied Statistics, Dedicated to A. Linder. Experentia Suppl. 22, 169–171.
Bhoj, D., and M. Ahsanullah. 1993. Testing equality of coefficients of variation of two populations. Biometrical J., 35(3), 355–359.
Chang, F., T. Lei, and A. C. M. Wong. 2012. Improved likelihood inference on testing the difference of non centrality parameters of two independent non central t distributions with identical degrees of freedom. Commun. Stat. Simulation Comput., 41(3), 342–354.
Forkman, J. 2009. Estimator and tests for common coefficients of variation in normal distributions. Commun. Stat. Theory Methods, 38(2), 233–251.
Johnson, N. L., S. Kotz, and N. Balakrishnan. 1994. Continuous univariate distributions, vol. II, 2nd ed. New York, NY: Wiley.
Kim, J., and A. J. Hayter. 2008a. Testing the equality of the non-centrality parameters of two non-central t-distributions with identical degrees of freedom. Commun. Stat. Simulation Comput., 37(9), 1709–1717.
Kim, J., and A. J. Hayter. 2008b. Efficient confidence interval methodologies for the non-centrality parameter of a non-central t-distribution. Commun. Stat. Simulation Comput., 37(4), 660–678.
Ledoit, O., and M. Wolf. 2007. Robust performance hypothesis testing with the Sharpe ratio. J. Empir. Finance, 15, 850–859.
Lehmann, E. L. 1986. Testing statistical hypotheses, 2nd ed. New York, NY: Springer-Verlag.
Leung, P. L., and W. K. Wong. 2008. On testing the equality of several Sharpe ratios, with application on the evaluation of iShares. J. Risk, 10(3), 15–30.
Mahmoudvand, R., and H. Hassani. 2009. Two new confidence intervals for the coefficient of variation in a normal distribution. J. Appl. Stat., 36(4), 429–442.
McKay, A. T. 1931. The distribution of the estimated coefficient of variation. J. R. Stat. Society, 94(4), 564–567.
McKay, A. T. 1932. Distribution of the coefficient of variation and the extended “t” distribution. J. R. Stat. Society, 95(4), 695–698.
Miller, G. E., and C. J. Feltz. 1997. Asymptotic inference for coefficients of variation. Commun. Stat. Theory Methods, 26(3), 715–726.
Miwa, T. 1994. Statistical inference on non-centrality parameters and Taguchi’s SN ratios. Proceedings of International Conference, Statistics in Industry, Science and Technology, 66–71.
Miwa, T. 2004. A normalizing transformation of noncentral F variables with large noncentrality parameters. Compstat 2004 Symposium, 1497–1502. Physica-Verlag/Springer.
Nagata, Y., M. Miyakaya, and T. Yokozawa. 2003. A test of the equality of several SN ratios for the systems with dynamic characteristics. J. Jpn. Society Qual. Control, 33, 83–92.
Nairy, K. S., and K. A. Rao. 2003. Tests of coefficients of variation of normal population. Commun. Stat. Simulation Comput., 32(3), 641–661.
Opdyke, J. D. 2007. Comparing Sharpe ratios: So where are the p-values? J. Asset Manage., 8, 308–336.
Pearson, K. 1896. Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia. Philos. Trans. R. Society, A, 187, 253–318.
Shafer, N. J., and J. A. Sullivan. 1986. A simulation study of a test for the equality of the coefficients of variation. Commun. Stat. Simulation Comput., 15, 681–695.
Sharpe, W. F. 1966. Mutual fund performance. J. Business, 39, 119–138.
Taguchi, G. 1977. Design of experiments, 3rd ed. Tokyo, Japan: Maruzen.
Tsou, T. S. 2009. A robust score test for testing several coefficients of variation with unknown underlying distributions. Commun. Stat. Theory Methods, 38, 1350–1360.
Vangel, M. G. 1996. Confidence intervals for a normal coefficient of variation. Am. Stat., 50(1), 21–26.
Verrill, S., and R. A. Johnson. 2007. Confidence bounds and hypothesis tests for normal distribution coefficients of variation. Commun. Stat. Theory Methods, 36, 2187–2206.
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Hayter, A.J., Kim, J. Small-Sample Tests for the Equality of Two Normal Cumulative Probabilities, Coefficients of Variation, and Sharpe Ratios. J Stat Theory Pract 9, 23–36 (2015). https://doi.org/10.1080/15598608.2014.881762
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DOI: https://doi.org/10.1080/15598608.2014.881762