CONTEMPORARY MATHEMATICS

287

Algebraic Methods in Statistics and Probability

AMS Special Session on Algebraic Methods in Statistics April 8-9 I 2000

University of Notre Dame Notre Dame I Indiana

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

Editors

Selected Titles in This Series

287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001

286 Edward L. Green, Serkan Ho§ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001

285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001

284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001

283 Paul A. Milewski, Leslie M. Smith, Fabian Waleffe, and Esteban G. Tabak, Editors, Advances in wave interaction and turbulence, 2001

282 Arlan Ramsay and Jean Renault, Editors, Groupoids in analysis, geometry, and physics, 2001

281 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering II, 2001

280 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering I, 2001

279 Alejandro Adem, Gunnar Carlsson, and Ralph Cohen, Editors, Topology, geometry, and algebra: Interactions and new directions, 2001

278 Eric Todd Quinto, Leon Ehrenpreis, Adel Faridani, Fulton Gonzalez, and Eric Grinberg, Editors, Radon transforms and tomography, 2001

277 Luca Capogna and Loredana Lanzani, Editors, Harmonic analysis and boundary value problems, 2001

276 Emma Previato, Editor, Advances in algebraic geometry motivated by physics, 2001 275 Alfred G. Noel, Earl Barnes, and Sonya A. F. Stephens, Editors, Council for

African American researchers in the mathematical sciences: Volume III, 2001 274 Ken-ichi Maruyama and John W. Rutter, Editors, Groups of homotopy

self-equivalences and related topics, 2001 273 A. V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, and C. Vinsonhaler,

Editors, Abelian groups, rings and modules, 2001 272 Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki, Editors, Quadratic

forms and their applications, 2000 271 J.P. C. Greenlees, Robert R. Bruner, and Nicholas Kuhn, Editors, Homotopy

methods in algebraic topology, 2001 270 Jan Denef, Leonard Lipschitz, Thanases Pheidas, and Jan Van Geel, Editors,

Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000 269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations

and foliations in dynamics, geometry and topology, 2001 268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential

geometric methods in the control of partial differential equations, 2000 267 NicolB.s Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen

Schneider, Editors, New trends in Hopf algebra theory, 2000 266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and

analytic geometry, 2000 265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique:

Homotopy theory in the Swiss Alps, 2000 264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Efim

I. Zelmanov, Editors, Combinatorial and computational algebra, 2000 263 Yan Guo, Editor, Nonlinear wave equations, 2000

(Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/287

Algebraic Methods in Statistics and Probability

CoNTEMPORARY MATHEMATICS

287

Algebraic Methods in Statistics and Probability

AMS Special Session on Algebraic Methods in Statistics April 8-9 I 2000

University of Notre Dame Notre Dame I Indiana

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

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeTurck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume contains the proceedings of the AMS Special Session on Algebraic Methods and Statistics held at The University of Notre Dame, Notre Dame, Indiana on April 8-9, 2000.

2000 Mathematics Subject Classification. Primary 05B20, 60F05, 62A01, 62C10, 62H15; Secondary 13P10, 15A52, 20B30, 30E20, 62G08.

Library of Congress Cataloging-in-Publication Data AMS Special Session on Algebraic Methods and Statistics (2000 : University of Notre Dame)

Algebraic methods in statistics and probability : AMS Special Session on Algebraic Methods and Statistics, April 8-9, 2000, University of Notre Dame, Notre Dame, Indiana / Marlos A. G. Viana, Donald St. P. Richards, editors. ·

p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 287) Includes bibliographical references. ISBN 0-8218-2687-5 (alk. paper) 1. Mathematical statistics--Congresses. 2. Probabilities-Congresses. 3. Algebra-

Congresses. I. Viana, Marlos A. G. II. Richards, Donald St. P. III. Title. IV. Contempo-rary mathematics (American Mathematical Society) ; v. 287.

QA276.A1 A48 2000 519.5-dc21 2001045884

Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissien@ams. erg.

Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.)

© 2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at URL: http: I /www. ams. erg/

10 9 8 7 6 5 4 3 2 1 06 05 04 03 02 01

Contents

Preface ix

Simplicial Inference JOHN AITCHISON 1

A Note on Nyman's Equivalent Formulation of the Riemann Hypothesis JEAN-FRANQOIS BURNOL 23

On the Construction of Linear Orthogonal Arrays by Extension DOMINIQUE COLLOMBIER AND ASTRID JOURDAN 27

A Coordinate-free Approach to Multivariate Exponential Families ALESSANDRO DI BUCCHIANICO AND DANIEL E. LOEB 37

Best Invariant Predictive Distributions MORRIS L. EATON AND WILLIAM D. SUDDERTH 49

A Family of Probability Densities Related to the Riemann Zeta Function WERNER EHM 63

Local Field U-Statistics STEVEN N. EVANS 75

Krawtchouk Matrices from Classical and Quantum Random Walks PHILIP FEINSILVER AND JERZY KOCIK 83

Some Rank-based Hypothesis Tests for Covariance Structure and Conditional Independence

YONGHONG GAO AND JOHN I. MARDEN 97

Gaussian Measures as Limits on Irreducible Symmetric Spaces and Cones PIOTR GRACZYK 111

The Covariance Structure of the Multivariate Liouville Distributions RAMESHWAR D. GUPTA AND DONALD ST. P. RICHARDS 125

vii

viii CONTENTS

Reduction of Regression Models under Symmetry INGE S. HELLAND 139

Deconvolution Density Estimation on Compact Lie Groups PETER T. KIM AND DONALD ST. P. RICHARDS 155

On Efficiency of Indirect Estimation of Nonparametric Regression Functions CHRIS A. J. KLAASSEN, EuN-Joo LEE AND FRITS H. RuYMGAART 173

Patterned Matrices Treated via Linear Spaces TONU KOLLO AND DIETRICH VON ROSEN 185

Random Walks on Regular Languages and Algebraic Systems of Generating Functions

STEVEN P. LALLEY 201

The Normal Quasi-Wishart Distribution GERARD LETAC AND HELENE MASSAM 231

Rank Score Statistics for Spherical Data TERESA NEEMAN AND TED CHANG 241

Graphical Model Search via Essential Graphs MICHAEL D. PERLMAN 255

Computational Commutative Algebra in Discrete Statistics GIOVANNI PISTONE, EVA RICCOMAGNO AND HENRY P. WYNN 267

Maximum Covariance Difference Test for Equality of Two Covariance Matrices

AKIMICHI TAKEMURA AND SATOSHI KURIKI 283

The Covariance Structure of Random Permutation Matrices MARLOS A. G. VIANA 303

The Extendibility of Statistical Models ERNST WIT AND PETER McCuLLAGH 327

Preface

The role of algebraic methods and arguments in statistics and probability is well-known, from Gauss' least squares principle through R. A. Fisher's method of variance decomposition. The work of A. James [Jam57], according to E. Han-nan [Han65, p.65], appears to be among the first describing the group-theoretic nature of Fisher's argument, giving meaning to the notion of relationship algebra, or the commuting algebra of a representation of the group of symmetries of an experimental design. During the same period, U. Grenander [Gre63] showed the effectiveness of harmonic analysis techniques to extend the classical limit theorems to algebraic structures such as locally compact groups, Banach spaces and topo-logical algebras. Two decades later, the relevance of group invariance and group representation arguments in statistical inference would become evident in the works of, among others, P. Diaconis [Dia88], M. Eaton [Eat89], R. Farrell [Far85] and R. Wijsman [Wij90]. The integral ofHaar, as didactically presented in L. Nachbin's monograph [Nac65], became a familiar tool among statisticians, and S. Anders-son's group symmetry covariance models are recognized (e.g., [Per87]) now as a landmark concept extending and setting definitive boundaries to the standard of multivariate statistical analysis, as known in the tradition of T.W. Anderson. The contributions of S. Lauritzen [Lau96] on modelling complex stochastic systems us-ing graph theory, and the monograph by G. Pistone, E. Riccomagno and H. Winn [PRWOO] on computational commutative algebra in statistics are examples of re-cent new complements to the seminal work of Alan James.

The opportunity present to all of us statisticians and probabilists today is in the formulation and application of comprehensive concepts and language capable of capturing in their essence the research questions posed by those with whom we collaborate - reasoning tools which represent the broad physical, biological, behav-ioral, social interface between phenomena and data. The challenge is in representing and including the interface, the context, in our proposed methods of scientific expla-nation. As recently observed by V. Lakshminarayanan [LSJ98], group-theoretical analysis may lead to prediction of new measurable effects or allow newer methods of analysis of data and offer greater insight into [specific] research hypotheses. This remark exemplifies the many opportunities ahead.

The present volume was originally proposed by the editors during the AMS Sectional Meeting, in Montreal, 1997. A later sectional meeting in South Bend, Indiana, in the Spring of 2000, gave us the opportunity to continue with the project. The papers appearing in the South Bend volume result from presented talks and subsequent invited contributions.

We are grateful to all who assisted us during the refereeing process, namely J. Aitchison, S. Andersson, J-F. Burnol, P. Chan, T. Chang, D. Collombier,

ix

X PREFACE

D. Dawson, I. Dinwoodie, C. Dunkl, W. Ehm, S. Evans, J. Faraut, U. Franz, D. Geiger, S. Hedayat, I. Helland, T. Kollo, S. Lalley, H. Massam, B. McNeney, R. Mukerjee, H. Neudecker, M. Perlman, F.'Ruymgaart, and P. Sarnak. We also ac-knowledge the support of the Series managing editor, Dennis DeTurck. Our sincere appreciation goes to the authors, for their timely cooperation during the editorial process.

We feel that the twenty-three contributions that follow give an excellent over-view of current opportunities and directions in algebraic methods in statistics and probability.

[Dia88]

[Eat89]

[Far85] [Gre63] [Han65]

[Jam57]

[Lau96] [LSJ98J

[Nac65] [Per87]

[PRWOOJ

[Wij90J

References P. Diaconis, Group representation in probability and statistics, IMS, Hayward, Califor-nia, 1988. M. L. Eaton, Group invariance applications in statistics, IMS-ASA, Hayward, Califor-nia, 1989. R. H. Farrell (ed.), Multivariate calculation, Springer-Verlag, New York, NY, 1985. U. Grenander, Probabilities on algebraic structures, Wiley, New York, NY, 1963. E. J. Hannan, Group representations and applied probability, Journal of Applied Prob-ability 2 (1965), 1-68. A. T. James, The relationship algebra of an experimental design, Annals of Mathemat-ical Statistics 28 (1957), 993-1002. S. L. Lauritzen, Graphical models, Oxford University Press, New York, NY, 1996. V. Lakshminarayanan, R. Srudhar, and R. Jagannathan, Lie algebraic treatment of dioptric power and optical aberrations, J. Optical Society of America, A 15 (1998), 2497-2503. L. Nachbin, The Haar integral, Van Nostrand, Princeton, N.J., 1965. M. D. Perlman, Group symmetry covariance models, Statistical Science 2 (1987), 421-425. G. Pistone, E. Riccomagno, and P. Wyinn, Algebraic statistics: Computational com-mutative algebra in statistics, Chapman & Hall/CRC, Boca Raton, FL, 2000. R. A. Wijsman, Invariant measures on groups and their use in statistics, IMS, Hayward, California, 1990.

Marlos A. G. Viana Chicago, Illinois

Donald St. P. Richards Charlottesville, Virginia

July 26, 2001

Selected Titles in This Series (Continued from the front of this publication)

262 Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors, Crystallographic groups and their generalizations, 2000

261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability on algebraic structures, 2000

260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000

259 D. V. Huynh, S. K. Jain, and S. R. Lopez-Permouth, Editors, Algebra and its applications, 2000

258 Karsten Grove, Ib Henning Madsen, and Erik Kjoor Pedersen, Editors, Geometry and topology: Aarhus, 2000

257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000

256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000

255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear PDE's, dynamics and continuum physics, 2000

254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000

253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000

252 Nathaniel Dean, Cassandra M. McZeal, and Pamela J. Williams, Editors, African Americans in Mathematics II, 1999

251 Eric L. Grinberg, Shiferaw Berhanu, Marvin I. Knopp, Gerardo A. Mendoza, and Eric Todd Quinto, Editors, Analysis, geometry, number theory: The Mathematics of Leon Ehrenpreis, 2000

250 Robert H. Gilman, Editor, Groups, languages and geometry, 1999 249 Myung-Hwan Kim, JohnS. Hsia, Yoshiyuki Kitaoka, and Rainer

Schulze-Pillot, Editors, Integral quadratic forms and lattices, 1999 248 Naihuan Jing and Kailash C. Misra, Editors, Recent developments in quantum

affine algebras and related topics, 1999 247 Lawrence Wasson Baggett and David Royal Larson, Editors, The functional and

harmonic analysis of wavelets and frames, 1999 246 Marcy Barge and Krystyna Kuperberg, Editors, Geometry and topology in

dynamics, 1999 245 Michael D. Fried, Editor, Applications of curves over finite fields, 1999 244 Leovigildo Alonso Tarrio, Ana Jeremias Lopez, and Joseph Lipman, Studies in

duality on noetherian formal schemes and non-noetherian ordinary schemes, 1999 243 Tsit Yuan Lam and Andy R. Magid, Editors, Algebra, K-theory, groups, and

education, 1999 242 Bernheim Booss-Bavnbek and Krzysztof Wojciechowski, Editors, Geometric

aspects of partial differential equations, 1999 241 Piotr Pragacz, Michal Szurek, and Jaroslaw Wisniewski, Editors, Algebraic

geometry: Hirzebruch 70, 1999 240 Angel Carocca, Victor Gonzalez-Aguilera, and Rubi E. Rodriguez, Editors,

Complex geometry of groups, 1999

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej.

Algebraic Methods in Statistics and Probability Marlos A. G. Viana and Donald St. P. Richards, Editors

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 proba-bility. 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.

ISBN 0-8218-2687-5