Linear and Graphical Models: for the Multivariate Complex Normal Distribution
Springer Science & Business Media, May 19, 1995 - Mathematics - 183 pages
In the last decade, graphical models have become increasingly popular as a statistical tool. This book is the first which provides an account of graphical models for multivariate complex normal distributions. Beginning with an introduction to the multivariate complex normal distribution, the authors develop the marginal and conditional distributions of random vectors and matrices. Then they introduce complex MANOVA models and parameter estimation and hypothesis testing for these models. After introducing undirected graphs, they then develop the theory of complex normal graphical models including the maximum likelihood estimation of the concentration matrix and hypothesis testing of conditional independence.
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According to Theorem assume C+(G called characteristic function cliques Cnxp complex MANOVA model complex matrix complex normal distribution complex random matrix complex random variable complex random vector complex vector complex Wishart distributed conditional independence consider contains correspondence covariance CPXP decomposable decomposition deduce defined Definition denoted determined dimension distributed random element equal equivalent Example exists Figure Further let Furthermore let G-regular given Hence Hereby Hermitian holds hypothesis illustrated implies Let G Let X likelihood ratio test linear Markov property maximum likelihood estimate means multivariate complex normal mutually independent Note observe obtain operator orthogonal projection p-dimensional complex random partitioned positive probability Proof rank representing the orthogonal respectively separated simple undirected graph subset Theorem transformation univariate vector space w.r.t. Lebesgue measure zero