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

ISBN-10: 9048184398

ISBN-13: 9789048184392

ISBN-10: 9401706530

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**Extra resources for Advances in Multivariate Statistical Analysis: Pillai Memorial Volume**

**Example text**

But you need not find such a representation in order to be ready for applying the projection theorem. Sometimes quasi-inner products could be used as an alternative to convex analysis. 8 in La Motte (1982), p. 249, which is nothing else than the projection-theorem. 4. APPLICATIONS TO MATHEMATICAL PROGRAMMING. In this section we want to apply the projection theorem to find the minimizers of ~(x) = W(x,x), where W(x,y) =~x'By + ~ p'x + ~ p'y + d is a quasi-inner product on ffin. d. (non negative definite) matrix.

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. By the parallelogram-identity the latter expression is identical to W (x-y ,x-y) > 0.

A. Goodman, Academic Press, New York, pp. 529-543 . , and STUART, A. (1966): 'Data-dredging procedures in survey analysis'. The American Statistician, 20, no. 3, 20-23. H. (1985): 'Patterned covariances'. In: Encyclopedia of Statistical Sciences, vol. 6, eds. S. L. Johnson, Wiley, New York, pp. 638-641. Y. R. Krishnaiah and J. Schmidhammer EFFECT OF ADDITIONAL VARIABLES IN PRINCIPAL COMPONENT ANALYSIS, DISCRIMINANT ANALYSIS AND CANONICAL CORRELATION ANALYSIS 1. INTRODUCTION In a number of situations, it is of interest to find out whether the addition of a new set of variables gives additional information for inference.

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

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