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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H . DRYGAS 16 all x E V, x f 0, then W is called an inner product. (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.

16) and 2 W1(x,y1+y2) - j~1W1(x,yj) x+y x+y x+y +y ~ 2 ) + ¢(~)) - 4(¢(-Y) + ¢(y)) = 4(¢( - (~P(y1+y2) - 1P(Y1)- ¢(y2)). 17) By applying functional equation (i) to the first two summands, the second two summands and 1P(Y1 )+¢(y 2 ) we get Y1+y2 Y1+y2 2 W1(x,y1+y2)- j~1W1(x,yj) = -8¢(-4-) +4¢(~ H . 18) We now apply functional equation (ii) to k = 3, A1 = 1, ~ 1 = 1, ~ 2 = -6, ~ 3 = 8. Then it is readily A2 = ~. A3 = verified that ~ ~- lA· 12 = ~~-A· = 0. 18) must vanish, which proves 2 w1 (x,y 1 ,y 2 ) = j~ 1 w 1 (x,yj).

93 1 ) M. ochrogaster 2 female M. ochrogaster 2 male '¥3 188 . 79 : ) j '¥4 I (86 . 17 ! 1 of table II, but they have one exactly identical characteristic vector. S. Pillai , whom I admired as one of the greatest multivariate statisticians, and whose advice enabled me to do research into problems of relationship between covariance matrices . S. (1984): 'Age variation in voles (Microtus californicus , Microtus ochrogaster) and its significance for systematic studies'. Occasional Papers of the Museum of Natura ·!

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