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Lecture Notes Editorial Policies Lecture Notes in Statistics provides a format for the informal and quick publication of monographs, case studies, and workshops of theoretical or applied importance. Thus, in some instances, proofs may be merely outlined and results presented which will later be published in a different form. Publication of the Lecture Notes is intended as a ser vice to the international statistical community, in that a commercial publisher, Springer-Verlag, can provide efficient distribution of documents that would other wise have a restricted readership. Once published and copyrighted, they can be documented and discussed in the scientific literature. Lecture Notes are reprinted photographically from the copy delivered in camera-ready form by the author or editor. Springer-Verlag provides technical instructions for the preparation of manuscripts. Volumes should be no less than 100 pages and preferably no more than 400 pages. A subject index is expected for authored but not edited volumes. Proposals for volumes should be sent to one of the series editors or addressed to "Statistics Editor" at Springer-Verlag in New York. Authors of monographs receive 50 free copies of their book. Editors receive 50 free copies and are responsi ble for distributing them to contributors. Authors, edi tors, and contributors may purchase additional copies at the publisher's discount. No reprints of individual contributions will be supplied and no royalties are paid on Lecture Notes volumes. Springer-Verlag secures the copyright for each volume. Series Editors: Professor P. Bickel Department of Statistics University of California Berkeley, California 94720 USA Professor P. Diggle Department of Mathematics Lancaster University Lancaster LA I 4 YL England Professor S. Fienberg Department of Statistics Carnegie Mellon University Pittsburgh, Pennsylvania 15213 USA Professor K. Krickeberg 3 Rue de L'Estrapade 75005 Paris France Professor 1. 01kin Department of Statistics Stanford University Stanford, California 94305 USA Professor N. Wermuth Department of Psychology Johannes Gutenberg University Postfach 3980 D-6500 Mainz Germany Professor S. Zeger Department of Biostatistics The Johns Hopkins University 615 N. Wolfe Street Baltimore, Maryland 21205-2103 USA

Lecture Notes in Statistics Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg, I. OIkin, N. Wermuth, and S. Zeger 174

Springer Science+ Business Media, LLC

Yasuko Chikuse Statistics on Special Manifolds Springer

Yasuko Chikuse Faculty of Engineering Kagawa University 2217-20 Hayashi-cho Takamatsu, Kagawa Japan chikuse@eng.kagawa-u.ac.jp Library of Congress Cataloging-in-Publication Data Chikuse, Yasuko. Statistics on special manifolds / Yasuko Chikuse. p. cm. - (Lecture notes in statistics; 174) Includes bibliographical references and index. 1. Manifolds (Mathematics) 2. Mathematical statistics. I. Title. II. Lecture notes in statistics (Springer-Science+Business Media, LLC) ; 174. QA613 .C48 516.07 - dc21 2002 2002042668 ISBN 978-0-387-00160-9 DOI 10.1007/978-0-387-21540-2 ISBN 978-0-387-21540-2 (eBook) © 2003 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 2003 AU rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Science+ Business Media, LLC ), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987 654 3 2 1 Typesetting: Pages created by the author using a Springer T EX macro package.

To my parents

Preface The special manifolds of interest in this book are the Stiefel manifold and the Grassmann manifold. Formally, the Stiefel manifold Vk,m is the space of k frames in the m-dimensional real Euclidean space Rm, represented by the set of m x k matrices X such that X' X = Ik, where Ik is the k x k identity matrix, and the Grassmann manifold Gk,m-k is the space of k-planes (k-dimensional hyperplanes) in Rm. We see that the manifold Pk,m-k of m x m orthogonal projection matrices idempotent of rank k corresponds uniquely to Gk,m-k. This book is concerned with statistical analysis on the manifolds Vk,m and Pk,m-k as statistical sample spaces consisting of matrices. The discussion is carried out on the real spaces so that scalars, vectors, and matrices treated in this book are all real, unless explicitly stated otherwise. For the special case k = 1, the observations from V1,m and G1,m-l are regarded as directed vectors on a unit sphere and as undirected axes or lines, respectively. There exists a large literature of applications of directional statis tics and its statistical analysis, mostly occurring for m = 2 or 3 in practice, in the Earth (or Geological) Sciences, Astrophysics, Medicine, Biology, Meteo rology, Animal Behavior, and many other fields. Examples of observations on the general Grassmann manifold Gk,m-k arise in the signal processing of radar with m elements observing k targets. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on the Grassmann manifold, which is one of the purposes of this book, must make some contributions to the study in the related sciences. This book is designed as a reference book for both theoretical and applied statisticians. The book will also be used as a textbook for a graduate course in multivariate analysis for students who specialize in mathematical statistics or multivariate analysis. I may assume that the reader is familiar with the usual theory of univariate statistics and has a thorough background in mathematics, in particular, a knowledge of multivariate calculation techniques. To make the book self-contained, a brief review of those aspects and other related topics is given in Chapter 1 and Appendices A and B. The reader may already know the usual theory of multivariate analysis on the real Euclidean space and intend

viii Preface to deepen or broaden the research area to the one treated in this book, that is, statistics on special manifolds, which is not treated in general textbooks of multivariate analysis. Chapters 1 to 3 are concerned with fundamental statistical and mathemat ical properties of the special manifolds Vk,m and Pk,m-k. Chapter 1 presents fundamental material which may be helpful for reading the main part of the book: the backgrounds of the special manifolds, exam ples of our orientation statistics in practical problems, and some multivariate calculation techniques and matrix-variate distributions. Chapter 2 discusses population distributions, uniform and non-uniform, on the special manifolds. Among those distributions, the matrix Langevin dis tributions defined on the two manifolds will be used for most of the statistical analyses treated in later chapters. Chapter 2 also looks at a method to generate some families of non-uniform distributions, that is, the distributions of the ori entation and the orthogonal projection matrix of a random rectangular matrix, and further suggests some simulation methods for generating pseudo-random matrices on the manifolds. Chapter 3 deals with the decomposition of the special manifolds, deriv ing various types of decompositions (or transformations) of random matrices and the corresponding decompositions of the invariant measures constructed in Chapter 1 (or Jacobians of the transformations). The results are not only of theoretical interest in themselves, but they are also of practical use for solving some distributional and inferential problems. In Chapter 4, we treat some distributional problems. The decompositions obtained in Chapter 3 are used to derive various distributional results and to introduce general families of distributions on the special manifolds. We derive various sampling distributions for the sample matrix sums, which are sufficient statistics, taken from the matrix Langevin distributions on the two manifolds. The forms of the sampling distributions are expressed in the integral forms involving hypergeometric functions with matrix arguments, which seem to be intractable. The hypergeometric functions with matrix arguments involved in inferential and distributional problems will be asymptotically evaluated and asymptotic analyses will be carried out for three cases, that is, for large sample size, for large concentration, and for high dimension, in Chapters 6, 7, and 8, respectively. Chapter 5 develops the theory of the statistical inference on the parameters of the matrix Langevin distributions on the special manifolds. The problems of estimation and tests for hypotheses of the parameters are dealt with by the Fisher (profile) scoring methods. These solutions are given in terms of hypergeometric functions with matrix arguments and will be approximately

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