On Bartlett's Test and Lehmann's Test for Homogeneity of Variances

Abstract

The purpose of this paper is to compare a modified likelihood ratio test (Bartlett [2]) with the asymptotically UMP invariant test (Lehmann [8]) for testing homogeneity of variances of $k$ normal populations. We denote these tests by the "$M$ test" and "$L$ test," respectively. The $M$ test has been investigated by many authors, whereas the $L$ test has not. Fitting beta type distributions, Mahalanobis [9] and Nayer [11] computed the percentage points of $M$, when the numbers of observations in each sample are the same. Nayer's results were confirmed by Bishop and Nair [3], using the exact null distribution of $M$ in a form of infinite series derived by Nair [10]. Asymptotic series expansion of the null distribution of $M$ was obtained by Hartley [6], using Mellin inversion formula, from which tables for percentage points were calculated by Thompson and Merrington [16], without assuming the equality of $k$-sample sizes. Later in a more general formulation, Box [4] derived the asymptotic series expansions of the null distributions of many test statistics, including that of the $M$ test, by using the characteristic function. Recently Pearson [12] obtained some approximate powers of the $M$ test both by fitting a gamma type distribution to the inverse of the modified likelihood ratio statistic and by using the Monte Carlo method. No attempt was made, however, to investigate the asymptotic non-null distribution of $M$. Sugiura [18] has shown the limiting distribution of $M$ in multivariate case under fixed alternative hypothesis to be normal. In Section 2 of this paper we shall show that the $L$ test is not unbiased, though the $M$ test is known to be unbiased (Pitman [14], Sugiura and Nagao [19]). In Section 3, we shall derive the limiting distributions of $L$ and $M$ under sequences of alternative hypothesis with arbitrary rate of convergence to the null hypothesis as sample sizes tend to infinity. Limiting distributions are characterized by $\chi^2$, noncentral $\chi^2$, and normal distributions, according to the rate of convergence of the sequence. In Section 4, asymptotic expansion of the null distribution of $L$ is given in terms of $\chi^2$-distributions, and asymptotic formulas for the percentage points of $L$ and $M$ are obtained by using the general inverse expansion formula of Hill and Davis [7], with some numerical examples. In Section 5, asymptotic expansions of the non-null distributions of $L$ and $M$ under a fixed alternative hypothesis are derived in terms of normal distribution function and its derivatives, from which approximate powers are computed. It may be remarked that the limiting non-null distributions of $L$ and $M$ degenerate at the null hypothesis, by which asymptotic null distributions cannot be derived.