Some Extensions of the Multivariate t-Distribution and the Multivariate Generalization of the Distribution of the Regression Coefficient

TLDR

Abstract

If the components x 1 , x 2 ,…, x k of a vector X have a non-singular multivariate normal distribution having a null vector of means and variance-covariance matrix Σ= σ 2 , the matrix R=[ρ ij ] (where ρ ii = 1) is known in certain cases but σ 2 is unknown. If s 2 is an estimate of σ 2 based on ƒ degrees of freedom and is distributed independently of X, the distribution of the vector t=x/ s is known as the multivariate t-distribution. This distribution was first obtained by Dunnett and Sobel (6) and independently by Cornish (3). Dunnett, Sobel and Bechhofer(2) have discussed some practical applications of this distribution. Cornish (3) obtained this distribution while considering the pre-treatment to be given to certain types of replicated experiments. This distribution possesses some useful properties and makes it suitable as a basis for exact tests of significance in various problems, and Dunnett and Sobel (6), by providing tables of the probability integral, have taken the first step towards its use in practice. Cornish, in a later paper (4) considered the sampling distribution of statistics derived from the multivariate t-distribution and using this he obtained the well-known ((7), (8)) distribution of the sample regression coefficient of one variate with respect to another, when both have a bivariate normal distribution.