Symmetric Multivariate and Related Distributions

230 Pages·1990·6.674 MB·English

by  Kai-Tai Fang, Samuel Kotz, Kai Wang Ng (auth.)

MONOGRAPHS ON
ST AT ISTICS AND APPLIED PROBABILITY
General Editors
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Stochastic Population Models in Ecology and Epidemiology
M.S. Bartlett (1960)
2 Queues D.R. Cox and W.L. Smith (1961)
3 Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964)
4 The Statistical Analysis of Series of Events D.R. Cox and
P.A. W. Lewis (1966)
5 Population Genetics W.J. Ewens (1969)
6 Probability, Statistics and Time M.S. Bartleu (1975)
7 Statistical lnference S.D. Silvey (1975)
8 The Analysis of Contingency Tables B.S. Everitt (1977)
9 Multivariate Analysis in Behavioural Research A.E. Maxwell (1977)
10 Stochastic Abundance Models S. Engen (1978)
11 Some Basic Theory for Statistical Inference E.J.G. Pitman (1979)
12 Point Processes D.R. Cox and V. Isham (1980)
13 Identification of Outliers D.M. Hawkins (1980)
14 Optimal Design S.D. Silvey (1980)
15 Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981)
16 Classification A.D. Gordon (1981)
17 Distribution-free Statistical Methods J.S. Maritz (1981)
18 Residualsand Infiuence in Regression R.D. Cook and S. Weisberg (1982)
19 Applications of Queueing Theory G.F. Newell (1982)
20 Risk Theory, 3rd edition R.E. Beard, T. Pentikäinen and
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21 Analysis of Survival Data D.R. Cox and D. Oakes (1984)
22 An lntroduction to Latent Variable Models B.S. Everitt (1984)
23 Bandit Problems D.A. Berry and B. Fristedt (1985)
24 Stochastic ModeHing and Control M.H.A. Davis and R. Vinter (1985)
25 The Statistical Analysis of Compositional Data J. Aitchison (1986)
26 Density Estimation for Statistics and Data Analysis
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27 Regression Analysis with Applications G.B. Wetherill (1986)
28 Sequential Methods in Statistics, 3rd edition G.B. Wetherill (1986)
29 Tensor Methods in Statistics P. McCullagh (1987)
30 Transformation and Weighting in Regression J.R. Carroll and
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31 Asymptotic Techniques for Use in Statistics O.E. Barndorff-Nielsen
and D.R. Cox (1989)
32 Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989)
33 Analysis of Infectious Disease Data N.G. Becker (1989)
34 Design and Analysis of Cross-Over Trials B. Jones and
M.G. Kenward (1989)
35 Empirical Bayes Methods, 2nd edition J.S. Maritz and T. Lwin (1989)
36 Symmetrie Multivariate and Related Distributions K.-T. Fang,
S. Kotz and K.W. Ng (1989)
37 Generalized Linear Models, 2nd edition P. McCullagh and
J.A. Neider (1989)
38 Cyclic Designs J.A. John (1987)
39 Analog Estimation Methods in Econometrics C.F. Manski (1988)
(Full details concerning this series are available from the Publishers)
Symmetrie Multivariate
and Related Distributions
KAI-TAl FANG
Institute of Applied Mathematics, Academia Sinica, China
SAMUEL KOTZ
Department of Management Science and Statistics,
University of Maryland
KAIWANGNG
Department of Statistics, University of Hong Kong
Springer-Science+Business Media, B.V.
ISBN 978-0-412-31430-8 ISBN 978-1-4899-2937-2 (eBook)
DOI 10.1007/978-1-4899-2937-2
© 1990 K.-T. Fang, S. Kotz and K. W. Ng.
Originally published by Chapman and Hall in 1990.
Softcover reprint of the hardcover 1st edition 1990
Typeset in 10fl2pt Times by Thomson Press ( India) Ltd, New Delhi
All rights reserved. No part of this book may be reprinted or reproduced, or
utilized in any form or by any e/ectronic, mechanical or other means, now
known or hereafter invented, including photocopying and recording, or in any
information storage and retrieval system, without permission in writing from
the publisher.
British Library Cataloguing in Publication Data
Fang, Kai-Tang
Symmetrie multivariate and related distributions.
1. Multivariate analysis
I. Title. II. Kotz, Samuel. III. Ng, Kai W. IV. Series.
519.5'35
Library of Congress Cataloging in Publication Data
Fang, Kai-Tang.
Symmetrie multivariate and related distributions I Kai-Tang Fang,
Samuel Kotz, Kai W. Ng.
p. cm. --(Monographs on statistics and applied probability)
Bibliography: p.
Includes index.
1. Distribution (Probability theory) I. Kotz, Samuel. II. Ng,
Kai W. III. Title. IV. Series.
QA273.6.F36 1989
519.2'4-dc20 89-32953
CIP
To our wives: Tingmei, Rosalie and May for their Iove, constant
support, and understanding that sometimes symmetric multi
variate distributions may come first.
Contents
Preface ix
1 Preliminaries 1
1.1 Construction of symmetric multivariate distributions 1
1.2 Notation 10
1.3 Groups and invariance 14
1.4 Dirichlet distribution 16
Problems 24
2 Spherically and elliptically symmetric distributions 26
2.1 Introduction and definition 26
2.2 Marginal distributions, moments and density 33
2.3 The relationship between </J and F 38
2.4 Conditional distributions 39
2.5 Properties of elliptically symmetric distributions 42
2.6 Mixtures of normal distributions 48
2.7 Robust statistics and regression model 51
2.8 Log-elliptical and additive logistic elliptical distribu-
tions 55
2.9 Complex elliptically symmetric distributions 64
Problems 66
3 Some subclasses of elliptical distributions 69
3.1 Multiuniform distributions 70
3.2 Symmetrie Kotz type distributions 76
3.3 Symmetrie multivariate Pearson Type VII distribu-
tions 81
3.4 Symmetrie multivariate Pearson Type II distributions 89
3.5 Some other subclasses of elliptically symmetric distri-
butions 92
Problems 93
viii CONTENTS
4 Characterizations 96
4.1 Some characterizations of spherical distributions 96
4.2 Characterizations of uniformity 100
4.3 Characterizations of normality 105
Problems 110
5 Multivariate t -norm symmetric distributions 112
1
5.1 Definition of Ln 112
5.2 Some properties of Ln 114
5.3 Extended Tn family 122
5.4 Mixtures of exponential distributions 130
5.5 Independence, robustness and charaeterizations 134
Problems 139
6 Multivariate Liouville distributions 142
6.1 Definitions and properties 142
6.2 Examples 146
6.3 Marginal distributions 150
6.4 Conditional distribution 157
6.5 Charaeterizations 162
6.6 Seale-invariant statistics 169
6.7 Survival funetions 171
6.8 Inequalities and applications 174
Problems 179
7 «-Symmetrie distributions 181
7.1 oc-Symmetrie distributions 181
7.2 Decomposition of I-symmetrie distributions 185
7.3 Properties of I-symmetrie distributions 191
7.4 Some special cases of 'I' "(oc) 196
Problems 197
References 202
Index 213
Preface
This book represents the joint effort of three authors, located
thousands of miles apart from one another, and its preparation was
greatly facilitated by modern communication technology. lt may
serve as an additional example of international co-operation among
statisticians specializing in statistical distribution theory.
In essence we have attempted to amass and digest widely scattered
information on multivariate symmetric distributions which has
appeared in the Iiterature during the last two decades. Introductory
remarks at the beginning of each chapter summarize its content
and clarify the importance and applicability of the distributions
discussed in these chapters; it seems unnecessary, therefore, to dwell
in this Preface on the content of the volume.
It should be noted that this work was initiated by the first author,
who provided continuous impetus to the project, and a great many
of tbe results presented in tbe book stem from bis own researcb or
tbe research of bis associates and students during the last 15 years.
Some of tbe original contributions of tbe second author, wbo also
guided tbe Iiterature search, in tbe field of multivariate distributions
derived a decade ago bave beeo included in substantially extended
and generalized form. Contributions of the third author are equally
important in providing a coberent description of novel results
discussed in the last chapters oftbis volume, which incorporate some
of bis recent results on tbe topics, as weil as in co-ordinating the
whole project and shaping it into the typescript form. The authors
enjoyed working as a team and each one contributed to all the
chapters albeit in varying degrees.
Compilation of this work was preceded by painstaking and
comprehensive Iiterature search and study. Our most sincere thanks
are to anonymous librarians in many parts of the USA, England,
Europe, Peoples' Republic of China, Hong Kong and Canada, who
assisted us, far beyond the call of duty. Our tbanks arealso due to
X PREFACE
numerous authors (far too numerous to cite individually) who
generously supplied us with preprints and reprints of their recent
works in the area. Our special thanks are due to Sir David R. Cox,
who brought our work to the attention of Chapman and Hall
Publishing House and whose pioneering contributions in numerous
branches of statistics and distribution theory significantly inspired
our undertaking. We are thankful to Ms Elizabeth Johnston, Senior
Editor at Chapman and Hall, for her capable, experienced and patient
guidance related to editorial and technical matters and for facilitating
communication among the authors. We are grateful to the Chairman
of the Department of Statistics at the University of North Carolina,
Professor S. Cambanis (an important contributor to the field of
multivariate symmetric distributions), and Professor P.M. Bentler,
Department of Psychology, University of California at Los Angeles,
for the hospitality afforded the first author, and to York University,
Canada, for providing the third author with congenial facilities
during his one-year visit at the Department of Mathematics. Last but
not least, we are happy to acknowledge the skilful and efficient word
processing and typing service ofMrs Lucile Lo and Miss Irene Cheung.
The authors sincerely hope that their endeavour will stimulate
additional research in the field of multivariate symmetric distri
butions and will contribute to the rapid penetration of these
distributions into the field of applied multivariate analysis as a more
versatile alternative to the classical methods based on multivariate
normal distributions.
Regrettably due to constraints on the size of this book, problems
dealing with sampling from symmetric elliptical and related
distributions are not discussed in this volume. The authors hope to
tackle these important topics in a future publication.
Kai-Tai Fang
Institute of Applied Mathematics, Academia Sinica
P.O. Box 2734
100080 Beijing, China.
Samuel Kotz
Department of Management Science and Statistics
University of Maryland
College Park, Md. 20742, USA.
Kai W. Ng
Department of Statistics
University of Hong Kong, Hong Kong.