On the asymptotic non null distribution of some tests for homogeneity of variances and related results

Abstract

This paper presents a unified approach to find the asymptotic non null distribution of some statistics used to test the homoge-neity of variances of independent normal populations and the equality of scale parameters of independent gamma random variables. The asymptotic distribution of sample variance ratios under a sequence of local alternatives is first obtained and it is then used to derive the asymptotic distribution of a general class of test statistics, which includes the ones proposed by Bartlett (1937), Foster (1964) and Samiuddin (1978). All the tests in this class are proved to be asymptotically equivalent under a sequence of local alternatives. It is shown how the results for homogeneity of variance may be derived starting from the asymptotic distribution of a fixed number of independent gamma random variables divided by their sum, as the scale parameters tend to a common value and the shape parameters tend to infinity. Finally, this is used to find the asymptotic behavior of a class of tests for the equality of the scale parameters of several independent gamma random variables.