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AuthorMalley, James D. author
TitleStatistical Applications of Jordan Algebras [electronic resource] / by James D. Malley
ImprintNew York, NY : Springer New York, 1994
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Descript VII, 102 p. online resource


This monograph brings together my work in mathematical statistics as I have viewed it through the lens of Jordan algebras. Three technical domains are to be seen: applications to random quadratic forms (sums of squares), the investigation of algebraic simplifications of maxiยญ mum likelihood estimation of patterned covariance matrices, and a more wideยญ open mathematical exploration of the algebraic arena from which I have drawn the results used in the statistical problems just mentioned. Chapters 1, 2, and 4 present the statistical outcomes I have developed using the algebraic results that appear, for the most part, in Chapter 3. As a less daunting, yet quite efficient, point of entry into this material, one avoiding most of the abstract algebraic issues, the reader may use the first half of Chapter 4. Here I present a streamlined, but still fully rigorous, definition of a Jordan algebra (as it is used in that chapter) and its essential properties. These facts are then immediately applied to simplifying the M:-step of the EM algorithm for multivariate normal covariance matrix estimation, in the presence of linear constraints, and data missing completely at random. The results presented essentially resolve a practical statistical quest begun by Rubin and Szatrowski [1982], and continued, sometimes implicitly, by many others. After this, one could then return to Chapters 1 and 2 to see how I have attempted to generalize the work of Cochran, Rao, Mitra, and others, on important and useful properties of sums of squares


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

Mathematics Algebra Probabilities Mathematics Algebra Probability Theory and Stochastic Processes


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