Linear and Graphical Models

for the Multivariate Complex Normal Distribution

Heidi H. Andersen, Malene Hojbjerre, Dorte Sorensen, Poul S. Eriksen (Autoren)

Buch | Softcover

183 Seiten

1995 | Softcover reprint of the original 1st ed. 1995
Springer-Verlag New York Inc.
978-0-387-94521-7 (ISBN)

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In the last decade, graphical models have become increasingly popular as a statistical tool. 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.

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.

1 Prerequisites.- 1.1 Complex Matrix Algebra.- 1.2 A Vector Space Isomorphism.- 1.3 Complex Random Variables.- 1.4 Complex Random Vectors and Matrices.- 2 The Multivariate Complex Normal Distribution.- 2.1 The Univariate Complex Normal Distribution.- 2.2 The Multivariate Complex Normal Distribution.- 2.3 Independence, Marginal and Conditional Distributions.- 2.4 The Multivariate Complex Normal Distribution in Matrix Notation.- 3 The Complex Wishart Distribution and the Complex U-distribution.- 3.1 The Complex Wishart Distribution.- 3.2 The Complex U-distribution.- 4 Multivariate Linear Complex Normal Models.- 4.1 Complex MANOVA Models.- 4.2 Maximum Likelihood Estimation in Complex MANOVA Models.- 4.3 Hypothesis Testing in Complex MANOVA Models.- 5 Simple Undirected Graphs.- 6 Conditional Independence and Markov Properties.- 6.1 Conditional Independence.- 6.2 Markov Properties in Relation to Simple Undirected Graphs.- 7 Complex Normal Graphical Models.- 7.1 Notation.- 7.2 The Concentration Matrix.- 7.3 Complex Normal Graphical Models.- 7.4 Maximum Likelihood Estimation of the Concentration Matrix.- 7.5 Decomposition of the Estimation Problem.- 7.6 Hypothesis Testing in Complex Normal Graphical Models.- A Complex Matrices.- A.1 Complex Vector Space.- A.2 Basic Operations of Complex Matrices.- A.3 Inverse Matrix.- A.4 Determinant and Eigenvalues.- A.5 Trace and Rank.- A.6 Conjugate Transpose Matrix.- A.7 Hermitian Matrix.- A.8 Unitary Matrix.- A.9 Positive Semidefinite Complex Matrices.- A.10 Positive Definite Complex Matrices.- A.11 Direct Product.- A.12 Partitioned Complex Matrices.- B Orthogonal Projections.- Notation.

Reihe/Serie	Lecture Notes in Statistics ; 101
Zusatzinfo	183 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	0-387-94521-0 / 0387945210
ISBN-13	978-0-387-94521-7 / 9780387945217
Zustand	Neuware