Multivariate Statistical Analysis, Second Edition, by Narayan C. Giri PDF

By Narayan C. Giri

ISBN-10: 0824747135

ISBN-13: 9780824747138

ISBN-10: 0824757815

ISBN-13: 9780824757816

Read Online or Download Multivariate Statistical Analysis, Second Edition, (Statistics: a Series of Textbooks and Monographs) PDF

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Additional info for Multivariate Statistical Analysis, Second Edition, (Statistics: a Series of Textbooks and Monographs)

Example text

X @x1 @xp Let f ðxÞ ¼ x0 Ax where A ¼ ðaij Þ is a p Â p matrix. Since 8 P P aii x2i þ 2xi j=i aij xj þ k=i akl xk xl ; if A is symmetric; > > > l=i < P P 0 2 x Ax ¼ aii xi þ xi j=i aij xj þ xj j=i aji xi > > > þ Pk=i a x x ; if A is not symmetric: : kl k l l=i We obtain 8 Pp @f ðxÞ < 2 j¼1 aij xj ; if A is symmetric; ¼ P : p a x þ Pp a x ; if A is not symmetric: @xi j¼1 ij j j¼1 ji j 22 Chapter 1 Hence @f ðxÞ ¼ @x ( 2Ax if A is symmetric; ðA þ A0 Þx if A is not symmetric: Let A ¼ ðaij Þ be aPmatrix of dimension p Â p.

17 Show that for any lower triangular matrix A the diagonal elements are its characteristic roots. 18 Show that any orthogonal transformation may be regarded as the change of axes about a ﬁxed origin. 19 Show that for any nonsingular matrix A of dimension p Â p and non-null p-vector x, x0 ðA þ xx0 ÞÀ1 x ¼ x0 AÀ1 x : 1 þ x0 AÀ1 x 20 Let A be a nonsingular matrix of dimension p Â p and let x; y be two non-null p-vectors. Show that ðA þ xy0 ÞÀ1 ¼ AÀ1 À ðAÀ1 xÞðy0 AÀ1 Þ 1 þ y0 AÀ1 x 21 Let X be a p Â q matrix and let S be a p Â p nonsingular matrix.

Xp ðx1 ; . . ; xq jxqþ1 ; . . ;Xp ðxq¼1 ; . . ;Xp ðxqþ1 ; . . ; xp Þ of Xqþ1 ; . . ; Xp is not zero. ;Xp ðx1 ; . . ; xq jxqþ1 ; . . ; xq Þ of X1 ; . . ; Xq given that Xqþ1 ¼ xqþ1 ; . . 3) by replacing the probability density functions by the corresponding mass functions. The mathematical expectation of a random matrix X 0 X11 B .. X¼@ . X1p 1 Á Á Á Xq1 .. C . A Á Á Á Xqp of dimension p Â q (the components Xij are random variables) is deﬁned by 0 EðX11 Þ Á Á Á B .. EðXÞ ¼ @ . EðX1p Þ Á Á Á 1 EðXq1 Þ ..

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