# Algebraic Methods in Statistics and Probability

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*Marlos A. G. Viana; Donald St. P. Richards*

Algebraic methods and arguments in statistics and probability are well
known, from Gauss's least squares principle through Fisher's method of variance
decomposition. The relevance of group-theoretic arguments, for example, became
evident in the 1980s. Such techniques continue to be of interest today, along
with other developments, such as the use of graph theory in modelling complex
stochastic systems.

This volume is based on lectures presented at the AMS Special Session on
Algebraic Methods and Statistics held at the University of Notre Dame (Indiana)
and on contributed articles solicited for this volume. The articles are
intended to foster communication between representatives of the diverse
scientific areas in which these functions are utilized and to further the trend
of utilizing algebraic methods in the areas of statistics and probability.

This is one of few volumes devoted to the subject of algebraic methods in
statistics and probability. The wide range of topics covered in this volume
demonstrates the vigorous level of research and opportunities ongoing in these
areas.

#### Readership

Graduate students and research mathematicians interested in combinatorics, probability theory, stochastic processes, and statistics.

# Table of Contents

## Algebraic Methods in Statistics and Probability

- Contents vii8 free
- Preface ix10 free
- Simplicial Inference 112 free
- A Note on Nyman's Equivalent Formulation of the Riemann Hypothesis 2334
- On the Construction of Linear Orthogonal Arrays by Extension 2738
- A Coordinate-free Approach to Multivariate Exponential Families 3748
- Best Invariant Predictive Distributions 4960
- A Family of Probability Densities Related to the Riemann Zeta Function 6374
- Local Field U-Statistics 7586
- Krawtchouk Matrices from Classical and Quantum Random Walks 8394
- Some Rank-based Hypothesis Tests for Covariance Structure and Conditional Independence 97108
- Gaussian Measures as Limits on Irreducible Symmetric Spaces and Cones 111122
- The Covariance Structure of the Multivariate Liouville Distributions 125136
- Reduction of Regression Models under Symmetry 139150
- Deconvolution Density Estimation on Compact Lie Groups 155166
- On Efficiency of Indirect Estimation of Nonparametric Regression Functions 173184
- Patterned Matrices Treated via Linear Spaces 185196
- Random Walks on Regular Languages and Algebraic Systems of Generating Functions 201212
- The Normal Quasi-Wishart Distribution 231242
- Rank Score Statistics for Spherical Data 241252
- Graphical Model Search via Essential Graphs 255266
- Computational Commutative Algebra in Discrete Statistics 267278
- Maximum Covariance Difference Test for Equality of Two Covariance Matrices 283294
- The Covariance Structure of Random Permutation Matrices 303314
- The Extendibility of Statistical Models 327338