A NOTE ON BOX'S GENERAL METHOD OF APPROXIMATION FOR THE NULL DISTRIBUTIONS OF LIKELIHOOD CRITERIA 1

: Many multivariate test statistics (such as the likelihood ratio test for MANOVA) have null distributions whose moments are proportional to ratios of products of gamma functions. For any random variable W, 0 < or = W < or = 1, whose moments have the above-mentioned form, Box proposed an asymptotic expansion for the c. d. f. of W, which provides an accurate method for determining the critical constants defining rejection regions for the multivariate tests mentioned above. Although the method is useful, the calculations needed to obtain the coefficients in each asymptotic expansion must be obtained ab initio, and almost always involve cumbersome algebraic manipulations. In the present study, simplified algorithms are given for calculating the coefficients of the asymptotic expansion in the general case. In a certain special case (which includes the null distributions of the likelihood ratio criteria for MANOVA and for testing the independence among sets of variates), explicit formulas are derived for these coefficients. (Modified author abstract)