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By Ishiguro M., Sakamoto Y.

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The variables r and T are independent and the distribution of X is characterized by the distribution of r , and it is easily shown that T, for all X, is uniformly distributed on Sm=(xERm;x'x=I}, the unit sphere in R". 6 will show. In the proof, due to Kariya and Eaton (1977) and Eaton (1977) we will use the fact that the uniform , is the unique distribution on S,,,which is invariant under distribution on S orthogonal transformations. 4). 6. If X has an m-variate spherical distribution with P(X= 0)=0 and r =tlXll=(XX)'/2, T(X)=IIXlt-'X, then T(X) is uniformly distributed on S,, and T(X) and r are independent.

Now, the characteristic function of t’Y,,, where t E R”, is 3, f N ( at), = E[exp(iat‘yN) considered as a function of a € R’. Also I6 The Multivuriure Normul and Reluted Distributions . and since t'X, - t'p, t'X, -t'p,. is a sequence of independent and identically distributed random variables with zero mean and variance t'Zt, it follows by the univariate central limit theorem that, as N -,00, the asymptotic distribution of t'YN is N(0,t'2t) and, hence, as N --, 00 jN(a,t)-+exp(- fa*t'Ct) for all t and a, where the right side is the characteristic function of the N(0, t'Zt) distribution.

However, the calculations are fairly straightforward when sampling from a N,,,(p, Z) distribution. 6). 6. 3. T H E NONCENTRAL x2 AND F DISTRIBUTIONS Many statistics of interest in multivariate analysis and elsewhere have noncentral x 2 and F distributions. ” Here we will review these two distributions, and this will afford us an opportunity to introduce some definitions and notation that will be used later. 1. The generalized hypergeometric function (or series) is where ( a ) k = u ( a + 1) - ( a + k - 1).

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