Table Of ContentMONOGRAPHS ON
STATISTICS AND APPLIED PROBABILITY
General Editors
D.R. Cox, D.V. Hinkley, D. Rubin and B.W. Silverman
1 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. Bartlett (1975)
7 Statistical Inference 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 (l98 1)
17 Distribution-free Statistical Methods J.S. Maritz (1981)
18 Residuals and Influence in Regression R.D. Cook and S. Weisberg (1982)
19 Applications of Queueing Theory G.F. Newel1 (1982)
02 ksiR Theory, 3rd edition R.E. Beard, T. Pentikainen dna
E. Pesonen )4891(
12 Analysis fo Survival Data D.R. Cox dna .D Oakes )4891(
22 nA Introduction to Latent Variable Models B.S. Everitt (1984)
32 Bandit Problems .A.D Berry dna B. Fristedt (1985)
42 Stochastic Modelling and Control M.H.A. sivaD dna R. Vinter (1985)
52 The Statistical Analysis fo Compositional Data J. Aitchison )6891(
62 Density Estimation for Statistics and Data Analysis
B. W. Silverman )6891(
72 Regression with Analysis Applications .B.G Wetherill (1986)
82 Sequential Methods ni Statistics, 3rd edition G.B. Wetherill (1986)
92 Tensor Methods in Statistics .P McCullagh (1987)
03 Transformation and Weighting in Regression J.R. Carroll dna
.D Ruppert )8891(
13 Asymptotic Techniques for Use ni Statistics O.E. Barndorff-Nielsen
dna D.R. Cox )9891(
23 Analysis fo Binary Data, 2nd edition D.R. Cox dna E.J. Snell(1989)
33 Analysis fo Infectious Disease Data N.G. Becker (1989)
43 Design and sisylanA fo Cross-Over Trials B. Jones dna
M.G. Kenward (1989)
53 Empirical seyaB Methods, 2nd edition J.S. Maritz dna T. Lwin )9891(
63 Symmetric Multivariate and Related Distributions K.-T. Fang,
S. Kotz dna K.W. Ng (1989)
73 Generalized Linear Models, 2nd edition .P McCullagh dna
J.A. Nelder )9891(
83 cilcyC Designs J.A. John (1987)
93 Analog Estimation Methods in Econometrics C.F. Manski (1988)
(Full detaiclosn cerningt his series are availabflreo m the Publishers)
cirtemmyS etairavitluM
dna detaleR snoitubirtsiD
KAI-TA1 FANG
Institute of Applied Mathematics, Academia Sinica, China
SAMUEL KOTZ
Department of Management Science and Statistics,
University of Maryland
KAI WANG NG
Department of Statistics, University of Hong Kong
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
First published 1990 by CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
Reissued 2018 by CRC Press
© 1990 by K.-T. Fang, S. Kotz and K. W. Ng.
CRC Press is an imprint of Taylor & Francis Group, an Informa business
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British Library Cataloguing in Publication Data
Fang, Kai-Tang
Symmetric multivariate and related distributions.
1. Multivariate analysis
I. Title. II. Kotz, Samuel. III. Ng, Kai W. IV. Series.
519.5’35
ISBN 0–412–31430–4
Library of Congress Cataloging-in-Publication Data
Fang, Kai-Tang.
Symmetric multivariate and related distributions / Kai-Tang Fang,
Samuel Kotz, Kai W. Ng.
p. cm. –– (Monographs on statistics and applied probability)
Bibliography: p.
Includes index.
ISBN 0–412–31430–4
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
A Library of Congress record exists under LC control number: 89032953
Publisher’s Note
The publisher has gone to great lengths to ensure the quality of this reprint but points out that some imperfections in the original
copies may be apparent.
Disclaimer
The publisher has made every effort to trace copyright holders and welcomes correspondence from those they have been unable to
contact.
ISBN 13: 978-1-315-89794-3 (hbk)
ISBN 13: 978-1-351-07704-0 (ebk)
Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the
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To our wives: Tingmei, Rosalie dna May for their love, constant
support, dna understanding that sometimes symmetric multi-
variate distributions may come first.
ContCents ontents
ecaferPPreface ix
1 1 seiranimilerPPreliminaries 1
11.1 .1 ConstrCuction onstruction foof multivariate distributions symmetric symmetric multivariate distributions 1
1.2 1.2 NotatioNn otation 10
3.11.3 GroGups roups and and invariance invariance 14
4.11.4 Dirichlet Ddistribution irichlet distribution 16
Problems Problems 24
2 2 yllacirehpSSpherically and and elliptically elliptically cirtemmyssymmetric distributions distributions 26
1.22.1 IntrodIuction and ntroduction and definition definition 26
2.2 2.2 Marginal Mdistributions, moments arginal distributions, moments and and density density 33
2.3 2.3 TheT he relationship relationship neewtebbetween 4 <P and and F F 38
2.4 2.4 ConditCional distributions onditional distributions 39
5.22.5 Properties Properties foof yllacitpilleelliptically symmetric symmetric distdributions istributions 42
2.6 2.6 Mixtures Mixtures foof normanl distributions ormal distributions 48
2.7 2.7 Robust Rstatistics obust statistics and and noissergerregression model model 51
8.22.8 Log-elliptical Log-elliptical and and distribu- elliptical additive logistic additive logistic elliptical distribu-
tions tions 55
9.22.9 Complex Complex yllacitpilleelliptically distributions symmetric symmetric distributions 64
Problems Problems 66
3 3 emoSSome s essalcbussubclasses foof elliptical elliptical distributions distributions 69
1.33.1 Multiuniform Mdistributions ultiuniform distributions 70
2.33.2 Symmetric Symmetric KKotz otz type type didstributions istributions 76
3.33.3 multivariate Symmetric SPearson Type ymmetric multivariate Pearson Type 11VVII distribu- distribu-
tions tions 81
4.33.4 multivariate Pearson Type Symmetric Symmetric multivariate Pearson Type I1 II didstributions istributions 89
5.33.5 emoSSome otoher ther sessalcbussubclasses foof yll acitpilleelliptically distri- symmetric symmetric distri-
butbions utions 92
Problems 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 characterizations 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 Characterizations 162
6.6 Scale-invariant statistics 169
6.7 Survival functions 171
6.8 Inequalities and applications 174
Problems 179
7 ex-Symmetric distributions 181
7.1 a-Symmetric distributions 181
7.2 Decomposition of !-symmetric distributions 185
7.3 Properties of !-symmetric distributions 191
7.4 Some special cases of 'P "( oc) 196
Problems 197
References 202
Index 213
Preface
This book represents the joint troffe fo three authors, located
thousands fo selim apart morf one anothearn,d its preparation saw
greatly facilitated yb modern communication technology. It yam
evres as an additional example fo international co-operation among
statisticians gnizilaiceps in statistical distribution theory.
In ecnesse ew evah attempted to amass and tsegid ylediw scattered
information on multivariate symmetric distributions hcihw has
appeared in the literature during the last owt decades. Introductory
remarks at the beginning fo each chapter summarize its content
and yfiralc the importance and applicability fo the distributions
ensieh tdessucsid it chapters; smees therefore, unnecessary, to llewd
ni this ecaferP on the content fo the .emulov
It should eb noted that this krow saw initiated yb the tsrif author,
ohw provided continuous impetus to the project, and a great many
the of results presented ni the book mets from sih nwo research or
eht research fo sih and associates studentsd uring the last 15 .sraey
emoS fo the original contributions fo the dnoces author, ohw also
dediug the literature search, ni the dleif fo multivariate distributions
devired a decade ago have peeb included ni substantially extended
and dezilareneg form. Contributions fo the third author are equally
important ni providing a coherent description fo levon results
dessucsid chapters last the in fo this ,emulov hcihw incorporate some
fo sih tnecer results on the topics, as llew as ni co-ordinating the
elohw project and shaping ti into the typescript form. The authors
deyojne working as a team and each one contributed to all the
chapters albeit ni varying .seerged
Compilation fo this krow saw dedecerp yb painstaking and
comprehensive literature search and study. Our most erecnis thanks
are to anonymous librarians in many parts fo the ,ASU England,
Europe, 'selpoeP Republic fo ChinaH,o ng Kong and Canada, ohw
detsissa ,su far beyond the llac fo duty. Our thanks are also due to
