Statistical Applications of Jordan Algebras

ISBN-13: 9780387943411 / Angielski / Miękka / 1994 / 102 str.

James D. Malley

Opis

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."

Spis treści

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.