On the evaluation of the probability integral of the multivariate t-distribution

Abstract

In the bivariate case, we shall write gn(tl, t2; P12) for gn(tl, t2; P) and Gn(tl, t2; P12) for Gn(t1, t2; p)If the random variables xj,..., xp follow the multivariate normal law with means zero, common variance o.2 and a correlation matrix (pij) and if (ns2)/o-2 is an independent x2 variable with n degrees of freedom, the random vector t = (tl, ..., tp), where ti = xi/s (i = 1, .. ., p) will have g,(t1,..., tp; P) as density function. Bechhofer, Dunnett & Sobel (1954) consider this distribution in connexion with a problem in the ranking of means of normal populations. Dunnett & Sobel (1954) give a formula for evaluating the probability integral when p = 2. Using this, they have prepared tables of the function Gn(h, h; ? 0.5) and its inverse. Dunnett & Sobel (1955) provide approximations for Gn(hl, ..., hp; P), valid when P satisfies certain conditions. In this paper we give an alternative formula for the evaluation of the probability integral. Though we too discuss only the bivariate case in detail, our method is of wider applicabilitv in the sense that it can be adopted to get the probability integral of the multivariate t-distribution of any dimension. We must mention here that the method of this paper is similar to that of Kendall (194 1) for evaluating the probability integral of the multivari'ate normal distribution. WVe have already mentioned that the multivariate t-distribution arises in the ranking of normal populations according to their means. We give more applications tow ards the end of this paper. It is shown how the multivariate t-distril)bition can be used in setting up simultaneous confidence bounds for the means of correlated( normal variables. Other applications are in constructing simultaneous confidence bounll(s for the parameters in a linear model and for future observations from a multivariate normnal distribution. Further applications will appear in a later paper.