Multivariate Statistics and Probability. Essays in Memory of Paruchuri R. Krishnaiah

574 Pages·1989·23.223 MB·English

by  C. R. Rao, M. M. Rao

P. R. Krishnaiah
1932-1987
The articles in this volume are dedicated to the memory of
the founding editor, P. R. Krishnaiah
Multivariate Statistics
and
Probability
Essays in Memory
of
Paruchuri R. Krishnaiah
Edited by
C.R. Rao
Center for Multivariate Analysis
Pennsylvania State University
University Park, Pennsylvania
M.M. Rao
Department of Mathematics
University of California
Riverside, California
®
ACADEMIC PRESS, INC.
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United Kingdom Edition published by
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Library of Congress Cataloging-in-Publication Data
Multivariate statistics and probability.
Includes bibliography and index.
1. Multivariate analysis. 2. Probabilities.
3. Krishnaiah, Paruchuri R. I. Krishnaiah,
Paruchuri R. II. Rao, C. Radhakrishna (Calyampudi
Radhakrishna), Date- . III. Rao, M.M. (Malempati
Madhusudana), Date-
QA278.M863 1989 519.535 89-14868
ISBN 0-12-580205-6 (alk. paper)
Printed in the United States of America
89 90 91 92 9 8 7 6 5 4 3 2 1
Contributors
Numbers in parentheses refer to the pages on which the authors' contributions begin.
G.J. Babu (15), Department of Statistics, Pennsylvania State University,
University Park, Pennsylvania 16802
Z.D. Bai (24, 40), Department of Mathematics, University of Pittsburgh,
Pittsburgh, Pennsylvania 15260
J.K. Baksalary (53), Academy of Agriculture, 60-769 Poznan, Poland
R.N. Bhattacharya (68, 80), Department of Mathematics, Indiana University,
Bloomington, Indiana 47405
H.W. Block (91), Department of Mathematics and Statistics, University of
Pittsburgh, Pittsburgh, Pennsylvania 15260
C.S. Chen (105), Department of Mathematics and Statistics, University of
Pittsburgh, Pittsburgh, Pennsylvania 15260
X.R. Chen (116), Department of Mathematics and Statistics, University of
Pittsburgh, Pittsburgh, Pennsylvania 15260
A. Cohen (131), Department of Statistics, Rutgers University, New Brunswick,
New Jersey 08903
M. Csörgd (151), Department of Mathematics and Statistics, Carleton
University, Ottawa, Ontario, Canada K1S5B6
CM. Cuadras (447), Department ofEstadistica, Universität de Barcelona,
Gram Via 585, 08007, Barcelona, Spain
M.H. DeGroot (539), Department of Statistics, Carnegie-Mellon University,
Pittsburgh, Pennsylvania 15213
R ErdÖS (169), Mathematical Institute, Hungarian Academy of Sciences,
Budapest, Hungary
D.A.S. Fraser (181), Department of Mathematics, York University, North York,
Ontario, Canada M3J1P3
Y. Fujikoshi (194), Department of Mathematics, Hiroshima University,
Hiroshima 730, Japan
ix
x Contributors
M. M. Gabr (484), Department of Mathematics, University of Manchester,
Institute of Science and Technology, Manchester M601QP, England
J.K. Ghosh (68), Indian Statistical Institute, Calcutta, India
M. Ghosh (206), Department of Statistics, University of Florida, Gainesville,
Florida 32611
N.C. Giri (270), Department of Mathematics and Statistics, University of
Montreal, Montreal PQ, Canada H3C3J7
P. Hall (228), Department of Statistics, Australian National University,
Cambera ACT 2601, Australia
T. Hida (225), Department of Mathematics, Nagoya University, Nagoya 464,
Japan
L. Horvath (151), Department of Mathematics, Szeged University, Szeged 6720,
Hungary
H.R Hucke (261), Department of Mathematics, Fern University, D-4040Neuss 1,
Federal Republic of Germany
G. Kallainpur (261), Department of Statistics, University of North Carolina,
Chapel Hill, North Carolina 27514
R.L. Karandikar (261), Indian Statistical Institute, New Delhi 110029, India
T. Kariya (270), Institute of Economic Research, Hitotubashi University,
Kunitachi Tokyo 186, Japan
S. Karl in (284), Department of Mathematics, Stanford University, Stanford,
California 94305
A.R. Karmous (300), Department of Statistics, Zagozig University, Zagozig,
Egypt
C.G. Khatri (319), Department of Statistics, Gujarat University, Ahmedabad
380009, India
T.L. Lai (334), Department of Statistics, Stanford University, Stanford,
California 94305
O. Lee (80), Department of Mathematics, Ewha Women's University, Seoul,
South Korea
K.-S. Lii (359), Department of Statistics, University of California, Riverside,
California 92521
M.E. Mack (539), Stuart Pharmaceutical Division ICI Americas, Inc.,
Wilmington, Delaware 19897
J.I. Marden (131), Department ofStatistics, University of Illinois, Champaign,
Illinois 61820
Contributors χι
T. Mathew (53), Department of Mathematics and Statistics, University of
Maryland, Catonsville, Maryland
B.Q. Miao (375), Department of Mathematics, University of Science and
Technology of China, Hefei, Peoples Republic of China
R. Nishii (392), Faculty of Integrated Arts and Sciences, Hiroshima University,
Hiroshima 730, Japan
F. Perron (270), Department of Mathematics and Statistics, University of
Montreal, Montreal PQ, Canada H3C3J7
J. PfanzagI (404), Mathematics Institute, Universität za Köln, DSOOOKöln 41,
Federal Republic of Germany
T.M. Pukkila (422), Department of Mathematical Sciences, University of
Tampere, SF-33101, Tampere, Finland
C.R. Rao (15, 24), Department of Statistics, Pennsylvania State University,
University Park, Pennsylvania 16802
M.B. Rao (505), Department of Mathematical Science, North Dakota State
University, Fargo, North Dakota 58105
M.M. Rao (343), Department of Mathematics, University of California,
Riverside, California 92521
P. Révész (169), Institut fur Statistics and Mathematics, Technische Universität
Wien, A-1040 Wien, Austria
Y. Rinott (284), Department of Statistics, Hebrew University, Jerusalem 91905,
Israel
M. Rosenblatt (359), Department of Mathematics, University of California,
San Diego, La Jolla, California 92093
C. Ruiz-Rivas (447), Department de Matematicas, Universidad Autonoma de
Madrid, Madrid, Spain
A.R. Samson (91), Department of Mathematics and Statistics, University of
Pittsburgh, Pittsburgh, Pennsylvania 15260
T.H. SavitS (105), Department of Mathematics and Statistics, University of
Pittsburgh, Pittsburgh, Pennsylvania 15260
V.V. Sazonov (457), Steklov Mathematical Institute, Academy of Sciences of the
USSR, Moscow B-333, USSR
RK. Sen (300), Department ofBiostatistics, School of Public Health,
University of North Carolina, Chapel Hill, North Carolina 27514
B.K. Sinha (206), Department of Mathematics and Statistics, University of
Maryland, Baltimore, Maryland 21228
xii Contributors
T. Subba Rao (484), Department of Mathematics, University of Manchester,
Institute of Science and Technology, Manchester M60IQD, England
K. Subramanyam (505), Department of Mathematics and Statistics, University
of Pittsburgh, Pittsburgh, Pennsylvania 15260
M. Tainguchi (521), Department of Mathematics, Hiroshima University,
Hiroshima 730, Japan
D.M. Titterington (228), Department of Statistics, University of Glasgow,
Glasgow G128QQ, United Kingdom
V.V. Ulyanov (457), Stecklov Mathematical Institute, Moscow B-333, USSR
J.S. Verducci (539), Department of Statistics, Ohio State University,
Columbus, Ohio 43210
Y.H. Wu (116), Department of Mathematics and Statistics, University of
Pittsburgh, Pittsburgh, Pennsylvania 15260
Z. Ying (334), Department of Statistics, University of Illinois, Champaign,
Illinois 61820
B.A. Zaieszkii (457), Stecklov Mathematical Institute, Moscow B-333, USSR
L.C. Zhao (24), Department of Mathematics, University of Science and
Technology of China, Hefei, Anhui, Peoples' Republic of China
Preface
The contributions included in the following pages were originally planned for a
presentation on the 55th birthday of Professor PR. Krishnaiah. Unfortunately his
illness became severe and soon after claimed his life. Hence this book is dedicated
to honor his memory. The articles, initially intended for a standard journal, are
based on original research by active and leading scientists in the areas of their cur-
rent interests in the multivariate field. The authors were all associated with
Krishnaiah professionally in his research and development of multivariate statisti-
cal analysis and stochastic theory, and many of them also knew him personally.
The chapters of this volume cover the main areas of multivariate statistical theory
and its applications, as well as aspects of probability and stochastic analysis. They
cover both finite sampling and asymptotic results, including aspects of decision
theory, Bayesian analysis, classical estimation, and regression, as well as time-
series problems. There are discussions of practical applications and computational
solutions. The works on probability include results on the (vector) central limit
theory for dependent random variables, the rates of convergence and asymptotic
expansions, Markov processes, and foundational problems. The material covered
in the volume should be of considerable interest to researchers as well as to gradu-
ate students working in theoretical and applied statistics, multivariate analysis, and
random processes.
We wish to express our appreciation to the contributors who responded to our
invitations and compiled their chapters within the time constraints. All the articles
were refereed; and, as a result, several underwent revisions and alterations. We are
deeply indebted to the many referees, whose names cannot be listed here, but
whose assistance was essential. Also our special thanks go to the staff of Academic
Press, Inc. for bringing out this volume expeditiously and accommodating all our
requests. Finally, we wish to acknowledge our home institutions for providing the
secretarial assistance to complete this project on schedule.
C.R. Rao
University Park, Pennsylvania
M.M. Rao
Riverside, California
May 1989
xiii
In Memoriam
P. R. Krishnaiah
(1932-1987)
Paruchuri Rama Krishnaiah was born in a suburb of Repalle in Andhra
Pardesh, India, in 1932 into a respected middle class Kisan (or farming)
family. He was one of the brightest students of the local high school and,
by his parents and teachers noticing this fact, he was sent to the well-
known Loyola College in Madras for higher education. Krishnaiah passed
the 2-year intermediate course in a high first class in 1950 and was admit-
ted to the then newly started and highly competitive (and difficult to gain
admission) statistics honors program in the Presidency College, also of
Madras University. Coincidentally, I joined the same school that year as a
(mathematics) graduate student (for a master's degree) and although we
saw each other, we met formally only at the end of the first quarter on a
trip home for a vacation. To our mutual surprise we found that we came
from adjacent districts, separated by the river Krishna, and our homes
were no more than 30 miles apart. From then on, we maintained a close
friendship, and thus it was a rude shock to me to learn in late 1986 that he
had become a victim of a cruel disease (cancer). Indeed he was a teetotaler
and a nonsmoker, had always been careful in whatever he ate, and had no
bad habits. He fought the ailment with great courage and was optimistic of
overcoming it, which he so expressed on his birthday, July 15, 1987.
Finally, he succumbed to the dreadful disease on August 1, 1987, leaving
his friends, loved ones, relatives, and colleagues in great sorrow. He is
survived by his wife, Indira, two young sons, Raghu and Niranjan, five
brothers, and two sisters. I shall now briefly describe his educational,
professional, organizational, humanitarian, and research accomplishments.
Soon after finishing his B. Sc. (Honors) at Presidency College, Madras,
in 1954, Krishnaiah went to the United States and joined the University of
Minnesota in Minneapolis to continue his graduate work in statistics. He
was always interested in both the theory and the applications of this sub-
ject. Even as a student he assisted in statistical methodology at the Bureau
of Educational Research in Minnesota, and this resulted in some
publications with his colleagues there. He took an M. S. degree in 1957
while searching for a suitable area of specialization for his dissertation.
1
Multivariate Statistics and Probability Reprinted from J. Mult. Anal. 27(1).
ISBN 0-12-580205-6 Copyright © 1988 by Academic Press, Inc.
All rights of reproduction in any form reserved.
2 IN MEMORIAM: P. R. KRISHNAIAH
During the summer term of 1956 the late Professor S. N. Roy of the
University of North Carolina visited Minnesota and offered a course in
multivariate statistical analysis. This was the first substantial account of
that subject given there; Krishnaiah took it and became interested in it
immediately. He spent the summer of 1957 at the IMS Summer Institute in
Boulder, Colorado, as a student member, where he was exposed to the
analysis of variance and related problems from the seminars of Professors
Bose, Kempthorne, Kruskal, Scheffé, and several other visiting scholars.
Another such session operated at the University of Minnesota a year later,
concentrating on ranking and selection problems, to which Krishnaiah was
again attracted. These three general areas of statistical theory became his
main research subjects for all his later work, consultation, and publications
as reviewed below.
There was no central location for statistics at Minnesota until 1960, and
students had to find the faculty whose interests (and appointments) were
combined with other areas. For a while, Krishnaiah traveled to discuss the
subjects of his interest, and in 1959-1960 he spent the year at Chapel Hill
with Professor Roy for this purpose. From 1960 on he worked as a senior
statistician at Remington Rand Univac in Blue Bell, Pennsylvania, until
1963 when he joined the Wright-Patterson Air Force Base in Dayton,
Ohio, as a mathematical statistician. He was also awarded the Ph. D.
degree by the University of Minnesota during the same year. Krishnaiah
remained at Wright-Patterson until 1976 when he joined the University of
Pittsburgh as a Professor of Statistics. In 1982 he became the founder and
director of the Center for Multivariate Analysis and also took a joint
appointment as a professor in the Graduate School of Business to reflect
his interests in substantiative applications. Before turning to his research, it
is appropriate to consider his professional and humanitarian work at this
point.
Krishnaiah organized six international symposia on Multivariate
Analysis which were held in June of 1965, 1968, 1972, 1975, and 1978 and
in July of 1983. It is of interest to note that he dedicated the published
"Proceedings" of the first conference to the memory of S. N. Roy and some
of the other proceedings volumes to H. Hotelling, P. C. Mahalanobis, and
H. Scheffé, paying his respects to these scholars from whose works and
contacts he had learned the subject. He also organized a symposium on
Applications of Statistics in June 1976, edited its proceedings, and gave
short courses on multivariate data analysis. He served as a member of the
council of the American Statistical Association for 1968-1969, was on
various committees of the IMS, received the Statistician of the Year award
in 1982 from the Pittsburgh chapter of the ASA, and was a member of the
technical committee on statistical pattern recognition of the International
Association of Pattern Recognition. Krishnaiah is a fellow of the ASA,