By D. J., Chambers, Joan Sybil Crawshaw

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**Read Online or Download A Concise Course in Advanced Level Statistics: With Worked Examples, Fourth Edition PDF**

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**Extra info for A Concise Course in Advanced Level Statistics: With Worked Examples, Fourth Edition**

**Example text**

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* ), with T: p x n, M: p x n, L: p x p, <1>: n x n, L 2: 0, 2: O. *