By A. K. Gupta, T. Varga (auth.)
In multivariate statistical research, elliptical distributions have lately supplied an alternative choice to the conventional version. lots of the paintings, although, is unfolded in journals in the course of the global and isn't simply obtainable to the investigators. Fang, Kotz, and Ng awarded a scientific research of multivariate elliptical distributions, even if, they didn't speak about the matrix variate case. lately Fang and Zhang have summarized the result of generalized multivariate research which come with vector in addition to the matrix variate distributions. nonetheless, Fang and Anderson accrued examine papers on matrix variate elliptical distributions, a lot of them released for the 1st time in English. They released very wealthy fabric at the subject, however the effects are given in paper shape which doesn't supply a unified remedy of the speculation. as a result, it appeared acceptable to assemble an important effects at the conception of matrix variate elliptically contoured distributions to be had within the literature and get them organized in a unified demeanour which could function an creation to the topic. The e-book should be important for researchers, academics, and graduate scholars in facts and comparable fields whose pursuits contain multivariate statistical research. elements of this publication have been awarded by means of Arjun okay Gupta as a one semester path at Bowling eco-friendly country collage. a few new effects have additionally been integrated which generalize the consequences in Fang and Zhang. wisdom of matrix algebra and records on the point of Anderson is thought. despite the fact that, bankruptcy 1 summarizes a few result of matrix algebra.
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Additional info for Elliptically Contoured Models in Statistics
PROOF: Since X = 0 iff tr(X'X) = 0, p(IIXII > 0) = I, follows so we can write X = Ilxll W· Define x = vec(X'). Then, x - Epn(o,Ipn,'II) and Ilxll = Ilxll· Hence, x = Ilxll F1r. x Let T(x) ="Flf' and G e O(pn). Then, we get Gx - Epn(o,Ipnll'll), so x == Gx and T(x) == T(Gx). On the other hand, Gx Gx T(Gx) = ~ = "Flf = GT(x) hence, T(x) '" GT(x). However, the uniform distribution is the only one on Spn which is invariant under orthogonal transformation. So, T(x) is uniformly distributed on Spn. l on Spn.
19) is equivalent to F(a) = 1. a) Therefore, h(a) =0 if and only if F(a) If a > 0 and F(a) < I, then for r ~ = 1. 0 we have 55 BASIC PROPERTIES 1 1 22 22 P(ra2 ::;; y) = P«r2 - a ) ::;; y IrO - rl) = a) 1 = peer ::;; (y2 + a 1 22 22 ) IrO - rl) = a). 17) we get 1 1 2 22 P(r::;; (y2 + a2) IrO - rl) = a) = 1. Take y = 0, then we get 2 Per 2 ::;; 0) = Per ::;; a IrO - rl)1/2 = a) = 1. 21) = 0) = 1. • In order to derive the conditional distribution for the multivariate elliptical distribution we need an additional lemma.
Let k(a) denote the p-dimensional vector whose a th element is 1 and all the others are 0 and l(b) denote the n:"dimensional vector whose b th element is 1 and all the others are O. Since X is nondegenerate, it must have an element xioio which is nondegenerate. 5) with t E R. Since, ! Oioio and i lPioio (2. = 1,2) are diagonal elements of positive semidefinite matrices, they cannot be negative, and since xioio is nondegenerate, they cannot be zero either. So, we can define 24 CHAPTER 2 Then, c > 0 and 'l'2(Z) = '1'1 (~Z) for Z E [0,00).
Elliptically Contoured Models in Statistics by A. K. Gupta, T. Varga (auth.)