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Improved tests for the equality of normal coefficients of variation

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The problem of testing the equality of coefficients of variation of independent normal populations is considered. For comparing two coefficients, we consider the signed-likelihood ratio test (SLRT) and propose a modified version of the SLRT, and a generalized test. Monte Carlo studies on the type I error rates of the tests indicate that the modified SLRT and the generalized test work satisfactorily even for very small samples, and they are comparable in terms of power. Generalized confidence intervals for the ratio of (or difference between) two coefficients of variation are also developed. A modified LRT for testing the equality of several coefficients of variation is also proposed and compared with an asymptotic test and a simulation-based small sample test. The proposed modified LRTs seem to be very satisfactory even for samples of size three. The methods are illustrated using two examples.

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  • Barndorff-Nielsen OE (1991) Modified signed log likelihood ratio. Biometrika 78:557–563

    Article  MATH  MathSciNet  Google Scholar 

  • Bhoj D, Ahsanullah M (1993) Testing equality of coefficients of variation of two populations. Biometric J 35:355–359

    Article  MathSciNet  Google Scholar 

  • DiCiccio TJ, Martin MA, Stern SE (2001) Simple and accurate one-sided inference from signed roots of likelihood ratios. Can J Stat 29:67–76

    Article  MATH  MathSciNet  Google Scholar 

  • Feltz GJ, Miller GE (1996) An asymptotic test for the equality of coefficients of variation from \(k\) normal populations. Stat Med 15:647–658

    Article  Google Scholar 

  • Forkman J (2009) Estimator and tests for common coefficients of variation in normal distributions. Commun Stat Theory Methods 38:233–251

    Article  MathSciNet  Google Scholar 

  • Fung WK, Tsang TS (1998) A simulation study comparing tests for the equality of coefficients of variation. Stat Med 17:2003–2014

    Article  Google Scholar 

  • Gerig TM, Sen AR (1980) MLE in two normal samples with equal but unknown population coefficients of variation. J Am Stat Assoc 75:704–708

    MathSciNet  Google Scholar 

  • Hamer AJ, Strachan JR, Black MM, Ibbotson C, Elson RA (1995) A new method of comparative bone strength measurement. J Med Eng Technol 19:1–5

    Article  Google Scholar 

  • Johnson NL, Welch BL (1940) Application of the nonecentral t-distribution. Biometrika 31:362–389

    MATH  MathSciNet  Google Scholar 

  • Krishnamoorthy K, Mathew T (2003) Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals. J Stat Plan Inference 115:103–121

    Article  MATH  MathSciNet  Google Scholar 

  • Lohrding RK (1969) A test of equality of two normal population means assuming homogeneous coefficient of variation. Ann Math Stat 40:1374–1385

    Article  MATH  MathSciNet  Google Scholar 

  • Miller EG, Karson MJ (1977) Testing the equality of two coefficients of variation. American Statistical Association: Proceedings of the Business and Economics Section, Part I, pp 278–283

  • Tian L (2005) Inferences on the common coefficient of variation. Stat Med 24:2213–2220

    Article  MathSciNet  Google Scholar 

  • Tian L, Wu J (2007) Inferences on the common mean of several log-normal populations: the generalized variable approach. Biometr J 49:944–951

    Article  MathSciNet  Google Scholar 

  • Vangel MG (1996) Confidence intervals for a normal coefficient of variation. Am Stat 50:21–26

    MathSciNet  Google Scholar 

  • Verrill, SP, Johnson RA (2007a) Confidence bounds and hypothesis tests for normal distribution coefficients of variation. United States Department of Agriculture, Forest Service, Research paper FPL-RP-638

  • Verrill S, Johnson RA (2007b) Confidence bounds and hypothesis tests for normal distribution coefficients of variation. Commun Stat Theory Methods 36:2187–2206

    Article  MATH  MathSciNet  Google Scholar 

  • Weerahandi S (2004) Generalized inference in repeated measures: exact methods in MANOVA and mixed models. Wiley, Hoboken, NJ

    Google Scholar 

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The authors are grateful to two reviewers for providing useful references and comments which improved the earlier version of the paper substantially.

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Correspondence to K. Krishnamoorthy.

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Krishnamoorthy, K., Lee, M. Improved tests for the equality of normal coefficients of variation. Comput Stat 29, 215–232 (2014).

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