Advanced Multivariate Statistics with Matrices
The book presents important tools and techniques for treating problems in m- ern multivariate statistics in a systematic way. The ambition is to indicate new directions as well as to present the classical part of multivariate statistical analysis in this framework. The book has been written for graduate students and statis- cians who are not afraid of matrix formalism. The goal is to provide them with a powerful toolkit for their research and to give necessary background and deeper knowledge for further studies in di?erent areas of multivariate statistics. It can also be useful for researchers in applied mathematics and for people working on data analysis and data mining who can ?nd useful methods and ideas for solving their problems. Ithasbeendesignedasatextbookforatwosemestergraduatecourseonmultiva- ate statistics. Such a course has been held at the Swedish Agricultural University in 2001/02. On the other hand, it can be used as material for series of shorter courses. In fact, Chapters 1 and 2 have been used for a graduate course ”Matrices in Statistics” at University of Tartu for the last few years, and Chapters 2 and 3 formed the material for the graduate course ”Multivariate Asymptotic Statistics” in spring 2002. An advanced course ”Multivariate Linear Models” may be based on Chapter 4. A lot of literature is available on multivariate statistical analysis written for di?- ent purposes and for people with di?erent interests, background and knowledge.
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applying approximation arbitrary assumption asymptotic distribution asymptotic normal basis vectors characteristic function column consider Corollary corresponding covariance cumulant function defined denoted density function differentiating eigenvalues eigenvectors elliptical distribution equals equation equivalent example exist expression follows from Theorem Fréchet derivative full rank Furthermore fw(W fy(x g-inverse given in Theorem Growth Curve model Hence Hermite polynomials holds idempotent implies inverse Lemma likelihood function matrix derivative matrix normal distribution maximum likelihood estimators MLNM MLNM(ABC moments Moreover multivariate normal multivariate normal distribution non-singular notation Note Np,n Observe obtained Orthogonal matrix p-vector parameters partitioned patterned matrix presented PROOF Proposition 1.2.2 random matrix random vector respectively sample dispersion matrix spherical distribution statement statistics subspaces Suppose symmetric matrix term Theorem transformation unique vecA Wishart density Wishart distribution Wo(X