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Statistical Applications of Jordan Algebras

Author: James D Malley
Publisher: New York, NY : Springer New York, 1994.
Series: Lecture notes in statistics (Springer-Verlag), 91.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
Summary:
This volume brings together the author's work in mathematical statistics as viewed through the lens of Jordan algebras. In particular the three main areas covered in this work are: applications to random quadratic forms (sums of squares); the investigation of algebraic simplifications of maximum likelihood estimation of patterned covariance matrices; and a more wide-ranging mathematical exploration of some of the  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: James D Malley
ISBN: 9781461226789 1461226783
OCLC Number: 853259084
Description: 1 online resource (vii, 102 pages).
Contents: 1 Introduction --
2 Jordan Algebras and the Mixed Linear Model --
2.1 Introductio --
2.2 Square Matrices and Jordan Algebra --
2.3 Idempotents and Identity Element --
2.4 Equivalent Definitions for a Jordan Algebr --
2.5 Jordan Algebras Derived from Real Symmetric Matrice --
2.6 The Algebraic Study of Random Quadratic Form --
2.7 The Statistical Study of Random Quadratic Form --
2.8 Covariance Matrices Restricted to a Convex Spac --
2.9 Applications to the General Linear Mixed Mode --
2.10 A Concluding Exampl --
3 Further Technical Results on Jordan Algebras --
3.0 Outline of this Chapte --
3.1 The JNW Theore --
3.2 The Classes of Simple, Formally Real, Special Jordan Algebra --
3.3 The Jordan and Associative Closures of Subsets of Sm --
3.4 Subspaces of --
3.5 Solutions of the Equation: sasbs 0 --
4 Jordan Algebras and the EM Algorithm --
4.1 Introductio --
4.2 The General Patterned Covariance Estimation Proble --
4.3 Precise State of the Proble --
4.4 The Key Idea of Rubin and Szatrowsk --
4.5 Outline of the Proposed Metho --
4.6 Preliminary Result --
4.7 Further Details of the Proposed Metho --
4.8 Estimation in the Presence of Missing Dat --
4.9 Some Conclusions about the General Solutio --
4.10 Special Cases of the Covariance Matrix Estimation Problem: Zero Constraint --
4.11 Embeddings for Constant Diagonal Symmetric Matrice --
4.12 Proof of the Embedding Problem for Sym(m)c --
4.13 The Question of Nuisance Parameter.
Series Title: Lecture notes in statistics (Springer-Verlag), 91.
Responsibility: by James D. Malley.
More information:

Abstract:

This volume brings together the author's work in mathematical statistics as viewed through the lens of Jordan algebras. In particular the three main areas covered in this work are: applications to random quadratic forms (sums of squares); the investigation of algebraic simplifications of maximum likelihood estimation of patterned covariance matrices; and a more wide-ranging mathematical exploration of some of the algebraic problems discussed. The author gives a full and rigorous definition of Jordan algebras and their essential properties and shows how they provide a natural and powerful algebraic tool for statisticians. In particular, the application of these methods to the M-step of the EM algorithm both simplifies this analysis and resolves some practical and important problems. This intertwining of ideas presented by the author will make this an interesting account suitable for researchers addressing these problems in statistics.

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