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Small-Sample Tests for the Equality of Two Normal Cumulative Probabilities, Coefficients of Variation, and Sharpe Ratios

  • A. J. HayterEmail author
  • Jongphil Kim
Article
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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.

Keywords

Cumulative distribution function Coefficient of variation Sharpe ratio Normal distribution Two-sample comparison Noncentral t-distribution Test procedure Acceptance set 

AMS Subject Classification

62F99 
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References

  1. 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.CrossRefGoogle Scholar
  2. 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.MathSciNetCrossRefGoogle Scholar
  3. 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.MathSciNetCrossRefGoogle Scholar
  4. 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.Google Scholar
  5. Bhoj, D., and M. Ahsanullah. 1993. Testing equality of coefficients of variation of two populations. Biometrical J., 35(3), 355–359.MathSciNetCrossRefGoogle Scholar
  6. 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.MathSciNetCrossRefGoogle Scholar
  7. Forkman, J. 2009. Estimator and tests for common coefficients of variation in normal distributions. Commun. Stat. Theory Methods, 38(2), 233–251.MathSciNetCrossRefGoogle Scholar
  8. Johnson, N. L., S. Kotz, and N. Balakrishnan. 1994. Continuous univariate distributions, vol. II, 2nd ed. New York, NY: Wiley.zbMATHGoogle Scholar
  9. 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.CrossRefGoogle Scholar
  10. 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.MathSciNetCrossRefGoogle Scholar
  11. Ledoit, O., and M. Wolf. 2007. Robust performance hypothesis testing with the Sharpe ratio. J. Empir. Finance, 15, 850–859.CrossRefGoogle Scholar
  12. Lehmann, E. L. 1986. Testing statistical hypotheses, 2nd ed. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
  13. 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.CrossRefGoogle Scholar
  14. 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.MathSciNetCrossRefGoogle Scholar
  15. McKay, A. T. 1931. The distribution of the estimated coefficient of variation. J. R. Stat. Society, 94(4), 564–567.CrossRefGoogle Scholar
  16. McKay, A. T. 1932. Distribution of the coefficient of variation and the extended “t” distribution. J. R. Stat. Society, 95(4), 695–698.CrossRefGoogle Scholar
  17. Miller, G. E., and C. J. Feltz. 1997. Asymptotic inference for coefficients of variation. Commun. Stat. Theory Methods, 26(3), 715–726.MathSciNetCrossRefGoogle Scholar
  18. 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.Google Scholar
  19. Miwa, T. 2004. A normalizing transformation of noncentral F variables with large noncentrality parameters. Compstat 2004 Symposium, 1497–1502. Physica-Verlag/Springer.Google Scholar
  20. 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.Google Scholar
  21. Nairy, K. S., and K. A. Rao. 2003. Tests of coefficients of variation of normal population. Commun. Stat. Simulation Comput., 32(3), 641–661.MathSciNetCrossRefGoogle Scholar
  22. Opdyke, J. D. 2007. Comparing Sharpe ratios: So where are the p-values? J. Asset Manage., 8, 308–336.CrossRefGoogle Scholar
  23. Pearson, K. 1896. Mathematical contributions to the theory of evolution. III. Regression, heredity and panmixia. Philos. Trans. R. Society, A, 187, 253–318.CrossRefGoogle Scholar
  24. 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.CrossRefGoogle Scholar
  25. Sharpe, W. F. 1966. Mutual fund performance. J. Business, 39, 119–138.CrossRefGoogle Scholar
  26. Taguchi, G. 1977. Design of experiments, 3rd ed. Tokyo, Japan: Maruzen.Google Scholar
  27. 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.MathSciNetCrossRefGoogle Scholar
  28. Vangel, M. G. 1996. Confidence intervals for a normal coefficient of variation. Am. Stat., 50(1), 21–26.MathSciNetGoogle Scholar
  29. 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.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  1. 1.Department of Business Information and AnalyticsUniversity of DenverDenverUSA
  2. 2.Department of BiostatisticsH. Lee Moffitt Cancer Center & Research InstituteTampaUSA

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