# Statistical Applications of Jordan Algebras

• James D. Malley
Book

Part of the Lecture Notes in Statistics book series (LNS, volume 91)

1. Front Matter
Pages i-vi
2. James D. Malley
Pages 1-4
3. James D. Malley
Pages 5-41
4. James D. Malley
Pages 42-62
5. James D. Malley
Pages 63-95
6. Back Matter
Pages 96-105

### Introduction

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.

### Keywords

Algebra Covariance matrix Likelihood SAS Variance expectation–maximization algorithm mathematical statistics

#### Authors and affiliations

• James D. Malley
• 1
1. 1.Division of Computer Research and TechnologyNational Institutes of HealthBethesdaUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-2678-9
• Copyright Information Springer-Verlag New York 1994
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-94341-1
• Online ISBN 978-1-4612-2678-9
• Series Print ISSN 0930-0325
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