By A. K. Gupta, T. Varga (auth.)
In multivariate statistical research, elliptical distributions have lately supplied a substitute for the traditional version. many of the paintings, even though, is opened up in journals through the global and isn't simply obtainable to the investigators. Fang, Kotz, and Ng provided a scientific learn of multivariate elliptical distributions, although, they didn't talk 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. however, Fang and Anderson accrued learn 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 offer a unified therapy of the idea. hence, it appeared acceptable to gather an important effects at the idea of matrix variate elliptically contoured distributions to be had within the literature and get them organized in a unified demeanour that may function an advent to the topic. The ebook could be valuable for researchers, lecturers, and graduate scholars in records and similar fields whose pursuits contain multivariate statistical research. elements of this publication have been awarded by way of Arjun okay Gupta as a one semester direction at Bowling eco-friendly nation collage. a few new effects have additionally been incorporated which generalize the implications in Fang and Zhang. wisdom of matrix algebra and facts on the point of Anderson is believed. although, bankruptcy 1 summarizes a few result of matrix algebra.
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Additional info for Elliptically Contoured Models in Statistics
Moreover, the relationship between g and c is given by rOO c= J ' - - - . f. f. of a pn-dimensional random vector y. f. of a pn-dimensional random vector y iff Jc g(y'y)dy = 1. 1, we get ~ 2 J c g(y'y)dy = c ~ rr r(T) d 00 pn l - g(r2)dr. jRpn Hence, we must have 00 and c = r(T) J -~-n-oo->--L-- . 3. c. distribution. 1. Let X - Ep,n(M,r. ® ,'lI), and partition X, M, and r. n is q x q, 1 ~ q < p. n ® ,'lI). PROOF: Let A = (Iq,O) be of dimensions q x p. e. • 32 CHAPTER 2 If we partition X vertically, we obtain the following result.
1, we get ~ 2 J c g(y'y)dy = c ~ rr r(T) d 00 pn l - g(r2)dr. jRpn Hence, we must have 00 and c = r(T) J -~-n-oo->--L-- . 3. c. distribution. 1. Let X - Ep,n(M,r. ® ,'lI), and partition X, M, and r. n is q x q, 1 ~ q < p. n ® ,'lI). PROOF: Let A = (Iq,O) be of dimensions q x p. e. • 32 CHAPTER 2 If we partition X vertically, we obtain the following result. 2. Let X - Ep,n(M,L ® ,'l'), and partition X, M, and as where Xl is p x m, Ml is p x m, and <1>11 is m x m, 1 ~ m < n. 3, it follows that X• = (x~)_ ((M~) (<1>11 <1>12) En,p , X 2 M " 2 <1>21 <1>22 ® L,'l' ) .
And from the continuity of the characteristic function and the fact that '1'2(0) = 1 it follows that 'l'2(U) = 0 for every u E lR. However, this is impossible. So, we must have 25 BASIC PROPERTIES L2 ® <1>2 = c(Ll ® <1>1). 16 it follows that there exist a > 0 and b > 0 such that L2 = aLl, <1>2 = b