An Application of a Generalized Gamma Distribution

Abstract

Let $\{x_{i1}, \cdots, x_{in_i}\}, i = 1, 2, \cdots, k, k \geqq 2, n_i \geqq 2$, be random samples from stochastically independent Gaussian populations with unknown means $\mu_i$ and unknown variances $\sigma^2_i$. To test the hypothesis $H_0:\sigma^2_1 = \cdots = \sigma^2_k = \sigma^2$, unknown, one may use a likelihood ratio test wherein $H_0$ is rejected if $\lambda < \lambda_0$, $\lambda = \prod^k_{i=1} \big\lbrack N \sum^{n_i}_{j=1} (x_{ij} - \bar{x}_i)^2/n_i \sum^k_{i=1}\sum^{n_i}_{j=1} (x_{ij} - \bar{x}_i)^2 \big\rbrack^{\frac{1}{2}n_i},$ $N = \sum^k_{i=1} n_i, \bar{x}_i = \sum^{n_i}_{j=1} x_{ij}/n_i$ and $\lambda_0$ is determined by the significance level. Under $H_0, Y_i = \sum^{n_i}_{j=1} (X_{ij} - \bar{X}_i)^2/\sigma^2$ are stochastically independent chi-square variables with $n_i - 1$ degrees of freedom (throughout capital letters denote random variables and the corresponding lower case letters values in their ranges). A general discussion of such derivations can be found in [2], pp. 183-195. The distribution of $\Lambda$ under $H_0$ is then obtainable from the extended form of the corollary below with $A_i = N/n_i, \beta_i = n_i/2, d_i = (n_i - 1)/2$. The corollary is a particularization of results obtained from a generalized gamma distribution, introduced by Stacy [3], with density \begin{equation*}\tag{1}f(y_i; a_i, d_i, p_i) = p_iy^{d_i-1}_i \exp \lbrack - (y_i/a_i)^{pi}\rbrack/a^{d_i}_i\Gamma(d_i/p_i)\end{equation*} for $y_i \geqq 0, a_i, d_i, p_i$ all positive. (Throughout, only the nonzero portions of densities will be indicated.) Inconveniently, formulas (4) and (6) below must be evaluated by numerical methods. Since the case $k > 3$ follows by induction from that for $k = 3$, only the cases $k = 2, 3$ will be given here.