International Journal of Scientific and Research Publications, Volume 4, Issue 2, February 2014 1 ISSN 2250-3153 www.ijsrp.org Tests for Equality of Coefficients of Variation of Two Normal Distributions for Correlated Samples Aruna Kalkur T. a , Aruna Rao K. b , St. Aloysius College, Mangalore. Karnataka, India. Mangalore University, Mangalagangothri. Karnataka, India. a Email: kalkuraruna@gmail.com Abstract: Coefficient of variation (C.V) is widely used as a measure of dispersion in applied research. C.V is unit less and thus facilitates the comparison of variability in two or more groups. Several tests have been proposed in the past for testing equality of C.Vs of two independent normal distributions. In plant sciences, medical sciences several characteristics of the plant or the subjects are to be compared regarding the variability and the samples are correlated. In this paper six tests are proposed for testing equality of C.Vs of a Bivariate normal distribution. The asymptotic null distribution of the entire test statistic is Chi-square distribution with 1 degree of freedom. The adequacy of the Chi-square approximation for finite samples is examined using simulation. Keywords:Coefficient of variation (C.V); Normal distribution; Bivariate normal distribution; Chi-square approximation; Simulation. I. INTRODUCTION Coefficient of Variation (C.V) is widely used as a measure of variation by the researchers in the applied disciplines like finance, climatology, engineering etc. The popularity of C.V stems from the fact that it is unit less and can be interpreted easily than standard deviation. C.V interpreted as relative risk in the area of finance [1]. In stock market analysis, it is interpreted as volatility per mean return and inverse C.V is referred as Sharp ratio [1]. Historically the first research work on C.V dates back to 1932 [2]. Initially, the researchers were interested to develop improved Confidence Interval for the C.V of the Normal Distribution. Resent references in these directions are Banik and Kibria [3] and see the references cited in this paper for earlier works. Although 100(1-α )% confidence interval at a level α test are interrelated, a formal Likelihood Ratio (LR) Test for equality of C.Vs of independent Normal Distributions was first introduced by Bennett [4] using modified C.V. Shafer and Sullivan [5] improved these tests using conditional likelihood. LR Test for equality of C.Vs of two independent Normal Distributions was proposed by Miller and Karson [1]. Doornbos and Dijkstra [6] extended the LR test for testing the equality of C.Vs of more than two independent Normal Distributions. Rao and Bhatta [7] proposed Wald test (also see [8]) for the same hypothesis. Gupta and Ma [9] proposed Score test for testing the equality of C.Vs. of two or more Normal Distributions. Following the generalized variable approach Tsui and Weerahandi [10], Jafari and Behboodian[11]developed generalized test statistic for testing the common C.V of two or more independent Normal Distributions. The LR, Wald and Score tests and their perturbed version were not robust against the assumption of normality. This has motivated Cabras, Mostallino and Racugno [12] to propose bootstrap tests for equality of C.Vs of two distributions. All these tests assumed that the samples are independent. In practice, correlated samples are often encountered. In medical studies many of the periodical characters are interrelated. For example, in the field of Anatomy, when gender has to be decided using the various measurements of the skull, these measurements are interrelated. This example is discussed in section 4. In the stock market the stock prices of various scripts are related and testing for equality of volatility for mean return for two or more scripts, the correlation needs to be accounted for Singh [13] proposed generalized test for testing equality of C.Vs of p variates Normal Distributions. This test is computationally tedious and it is not appealing to the scientists in the applied disciplines. To overcome this difficulty, we propose LR, Wald and Score tests for equaliy of C.Vs and Inverse Coefficient of Variations (ICV) from a Bivariate Normal Distribution. The finite sample performances of the tests are examined using