By Heidi H. Andersen, Malene Hojbjerre, Dorte Sorensen, Poul S. Eriksen
In the decade, graphical types became more and more well known as a statistical device. This ebook is the 1st which supplies an account of graphical versions for multivariate complicated general distributions. starting with an creation to the multivariate advanced general distribution, the authors increase the marginal and conditional distributions of random vectors and matrices. Then they introduce complicated MANOVA versions and parameter estimation and speculation checking out for those types. After introducing undirected graphs, they then boost the speculation of complicated common graphical versions together with the utmost probability estimation of the focus matrix and speculation trying out of conditional independence.
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Extra resources for Linear and Graphical Models: for the Multivariate Complex Normal Distribution
Example text
The word (n x p)-variate can be omitted if it is obvious from the context. 2 page 22. However it should not be inconvenient as we are only interested in arranging vectors in a matrix for n > 1. t. 4. The Multivariate Complex Normal Distribution in Matrix Notation 7r- np det (H)-n exp (- t 33 (Zj - Ojr H- 1 (Zj - OJ)) 3=1 ~hereS theorem. • ,Onr andz = (Z1,Z2, ... OJ) (Zj - OJ)" H- 1) ) , ,Znr E cnxp. 17 The density function ofCNnxp(S, In ® H) Let X be an n x p complex random matrix with £, (X) = CNnxp(S, In ® H), where S E cnxp and H E C~xp.
W) = C Wp (H, m), where H E C ~xp. JfV and Ware independent and n ~ p, then the distribution of U = det (V) det(V + W) is called a complex U-distribution with parameters p, m and n. This is denoted by £. (U) CU(p,m,n). = The complex U-distribution does not depend on H, which can be seen by the following considerations. The distributions of the independent complex matrices V and W are equal to the distributions of X' X and Y'Y, respectively, where X and Yare independent complex random matrices of dimensions n x p and m x p and with £.
W) = CW,,(H, n), where H E holds that C~xp. It E(W) =nH. Proof: Let W be a p x p complex random matrix with £. (W) = CWp(H,n), where H E Cr:,xp. Hence the distribution of W is also the distribution of X· X, where X is an n x p complex random matrix with £'(X) = CN'nx,,(O, In ® H). Writing X as X = (X b X 2 , ... ,X n)", where £. (Xj) = CN;,(O, H) we get E(W) n n j=1 ;=1 = E(X·X) = EE(XjX;) = EV(Xj ) = nH. • In the case where the dimension equals one the complex Wishart distribution becomes a chisquare distribution.