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Advances in Multivariate Statistical Analysis: Pillai by James O. Berger, Shun-Yu Chen (auth.), A. K. Gupta (eds.)

By James O. Berger, Shun-Yu Chen (auth.), A. K. Gupta (eds.)

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It may be noted that then W(x,y) is an inner product in the usual sense). 5 Theorem (a) Let W(x ,y) be a function defined on V x V, where V is a real or complex vector-space. , W 0 is a semi-inner (b) A non-negative quasi-inner product is always a quasiinner product. Proof W(ax+~y,ax+f3y) = a 2W(x,x) + a~(W(x,y)+W(y,x)) (a) if 0 ~a, f3 ~ + ~ 2 W(y,y), 1, a + f3 = 1. 5) if and only if a(1-a)[W(x,y)-2 Re W(x,y) + W(y,y)] > 0. If this holds for all a E [0,1] then necessarily W(x,x) -2 Re W(x,y) + W(y,y) > 0.

A > b for vectors, a,b-iff the relation holds for all-components of the two vectors). , Stoer-Witzgall (1970). The sufficiency simply follows from (Ax)'u = x'A'u ~ 0 if x,A'u ~ 0. d. 7) be given. 8) H . d. and symmetric. Since ~( x) = W(x,x) and C = {x : Ax~ b} is a conve~ ~et the projection(th)eorem 3. 1 applies. This yields that x~ 0 1 is optimal iff x 0 E C and W(x(o),x-x(o)) - W(x(ol,o) = ~(Bx(o)+p) 1 (x-x(o))::: 0"' x : Ax< b. 8). 9) is sufficient for optimality. Indeed (Bx(o)+p) (x-x(o)) = (-A U(o)) (x-x(o)) = (u(o)) Ax(o)- (u( 0 )) (Ax) = (u( 0 )) (b-y(o)) 1 1 1 1 1 - ( u( 0 )) Ax = ( u ( 0 )) 1 1 ( 1 b-Ax) ::: 0 , ( 4 .

An appropriate model could then be defined as follows. Level 4: The partial CPC model. For a fixed integer q < p-1, let '¥; = Ail~1f3l + ··· + A;ql3qf3~+ +A ~(i) ~(i)'+ + A j3(i)f3(i)' p ' ; ' q+1 q+ 1 q+ 1 . . 5) are the common characteristic vectors of all (i=1, ... ,k), where 13 1 to ~q and f3 (+i 1) to ~ ( i) are specific to each group. 5) that the characteristic vectors are ordered such that the common ones are labeled 1 to q. If we define the orthogonal matrices f3(i) as '¥. , 1 ~ (i) ·(i) (i) .

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