Exact distributions of Hotelling's generalized T20

Abstract

SUMMARY Percentile tables are given for the null distribution of Hotelling's generalized T2 statistic. These are obtained by analytic continuation of Constantine's series using a system of linear differential equations. The accuracy of certain approximations given by Pillai and Ito is discussed. T2= n2 tr (S1 S-1) = n2T, say, where S1 and S2 are independent m x m Wishart matrices on n1 and n2 degrees of freedom, respectively, estimating the same covariance matrix, with n2> >m. The exact null distribution when m = 2 was given by Hotelling in terms of the Gaussian hypergeometric function, and tabulated by F. E. Grubbs in 1954, in an unpublished report of Aberdeen Proving Ground, Maryland. In the general null case, the density function of T has been shown by the present author (1968) to satisfy a linear homogeneous differential equation of order m, with regular singularities at T = 0, -1, ..., - m and infinity. Constantine's (1966) series reduces in this case to the relevant solution of the differential equation in the unit circle about T = 0. Table 1 presents accurate percentiles of T2/n1 = (n2/n1) T, for m = 3 and 4 obtained by using an equivalent system of first order differential equations to carry out an analytic continuation of Constantine's series along the positive real axis. The accuracy of the procedure was checked by mapping the differential equation on to the unit interval using the change of variable Y = T/(T + 1). Agreement was generally to at least five significant figures. A similar tabulation has been carried out for m = 5.