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 maxi mum likelihood estimation of patterned covariance matrices, and a more wide 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.

Klappentext

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.

Inhalt
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.