By Samuel Kotz

First-class finished monograph on multivariate t distributions, with quite a few references. this can be the single publication focusing completely in this subject that i am acutely aware of.

similar in caliber and intensity to the "discrete/continuous univariate/multivariate distributions" sequence through Samuel Kotz, N. Balakrishnan, and Norman L. Johnson.

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Additional info for Multivariate T-Distributions and Their Applications

Sample text

28); specifically, T(X;E) = H(X;E)-H(X;D), where D is a diagonal matrix corresponding to E with the elements au,... , app This is due to the well known fact that uncorrelatedness implies independence in the normal case. This fact also implies that T(X; I) = 0. In general, for any member of an elliptical family of distributions, this is not true; in other words, uncorrelatedness does not imply that T(X) = 0. The mutual information attempts to summarize in a single number the whole dependence structure of the multivariate distribution of X.

K and RI,, is pi x p,,,, for l = 1, 2, ... , k, m = 1, 2, ... , k. Clearly P1 +P2 + +Pk = p. 22) = .. R11 R12 R21 R22 ... R11 R12 ... 25) Siotani showed that Yl = X1 and 1X(1)1- Y1+1 = v ql v X 1/2 (1 + vX (l) R, (x(l+l) - R(1+ 1)T R_1X(l) I for 1 = 1, ... (1) for 1 = 1, ... , k - 1. In the special case for R = Ir, the Y's can be written as Yl = X1 and y+q! Y1+1 Vv l + 1v E Xm,Xm -1/2 X1+1. 12 Quadratic Forms If X has the p-variate t distribution with degrees of freedom v, mean vec- tor µ, and correlation matrix R, then XTR-1X/p has the noncentral F distribution with degrees of freedom p and v and noncentrality parameter µT R-l t/p.

28) log(2e7r) + f log JR1. 27) will be discussed in Chapter 9. 29), one can obtain the entropy of the noncentral pvariate Cauchy distribution. 29) coincides with the entropies for the univariate Student's t and Cauchy distributions given, for example, in Lazo and Rathie (1978). 1). 26) subject to the constraints that define the class of pdfs considered. 1) is the solution to maximizing E[- log f (X)] subject to the constraint E [log { 1 + 1 (X - µ)T R-1 (X l v /p+V; p wf 2 2 ' )J where w(x; a) = 0(x) - 0(x - a), x > a, and denotes the digamma function.