| Abstract | Statistics that are the ratio of two independent linear combinations of independent chi-squared random variables are used to test hypotheses on parameters in mixed and random-effects models and to test hypotheses on the parameters in the joint analysis model of several experiments. We will call these statistics generalized F statistics and the associated test a generalized F test. In this paper we first obtain the exact distribution of any statistic that is the ratio of two independent linear combinations of independent Gamma distributed random variables. Based on this distribution we then obtain asymptotic and near-exact distributions for such statistics. Then, the exact, asymptotic and near-exact distributions of generalized F statistics are readily derived, under both the null andthe alternative hypotheses. Given that the exact distributions are infinite mixtures, they are not much adequate for practical purposes and thus the development of near-exact distributions is a desirable goal. Some examples of application are shown. |
