The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 40, Number 6 (1969), 2018-2032.
On Bartlett's Test and Lehmann's Test for Homogeneity of Variances
The purpose of this paper is to compare a modified likelihood ratio test (Bartlett ) with the asymptotically UMP invariant test (Lehmann ) 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  and Nayer  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 , using the exact null distribution of $M$ in a form of infinite series derived by Nair . Asymptotic series expansion of the null distribution of $M$ was obtained by Hartley , using Mellin inversion formula, from which tables for percentage points were calculated by Thompson and Merrington , without assuming the equality of $k$-sample sizes. Later in a more general formulation, Box  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  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  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 , Sugiura and Nagao ). 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 , 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.
Ann. Math. Statist., Volume 40, Number 6 (1969), 2018-2032.
First available in Project Euclid: 27 April 2007
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Sugiura, Nariaki; Nagao, Hisao. On Bartlett's Test and Lehmann's Test for Homogeneity of Variances. Ann. Math. Statist. 40 (1969), no. 6, 2018--2032. doi:10.1214/aoms/1177697282. https://projecteuclid.org/euclid.aoms/1177697282
- Acknowledgment of Prior Result: Nariaki Sugiura, Hisao Nagao. Acknowledgment of Priority to "On Bartlett's Test and Lehmann's Test for Homogeneity of Variances". Ann. Math. Statist., Volume 41, Number 4 (1970), 1373--1373.