Arjun K. Gupta – författare

A milestone in the published literature on the subject, this first-ever Handbook of Beta Distribution and Its Applications clearly enumerates the properties of beta distributions and related mathematical notions. It summarizes modern applications in a variety of fields, reviews up-and-coming progress from the front lines of statistical research and practice, and demonstrates the applicability of beta distributions in fields such as economics, quality control, soil science, and biomedicine. The book discusses the centrality of beta distributions in Bayesian inference, the beta-binomial model and applications of the beta-binomial distribution, and applications of Dirichlet integrals.

This volume is a collection of exercises with their solutions in Design and Analysis of Experiments. At present there is not a single book which collects such exercises. Theseexercises have been collected by the authors during the last four decadesduring their student and teaching years. They should prove useful to graduate students and research workers in Statistics. In Chapter I, theoretical results that are needed for understanding the material in this book, are given. Chapter 2 lists the exercises which have been collected by the authors. The solutions of these problems are given in Chapter 3. Finally an index is provided for quick reference. Grateful appreciation for financial support for Dr. Kabe's research at St. Mary's University is extended to National Research Council of Canada and St. May's Uni­ versity Senate Research Committee. For his visit to the Department ofMathematics and Statistics the authors are thankful to the Bowling Green State University.

This volume is a collection of exercises with their solutions in Design and Analysis of Experiments. At present there is not a single book which collects such exercises. Theseexercises have been collected by the authors during the last four decadesduring their student and teaching years. They should prove useful to graduate students and research workers in Statistics. In Chapter I, theoretical results that are needed for understanding the material in this book, are given. Chapter 2 lists the exercises which have been collected by the authors. The solutions of these problems are given in Chapter 3. Finally an index is provided for quick reference. Grateful appreciation for financial support for Dr. Kabe''s research at St. Mary''s University is extended to National Research Council of Canada and St. May''s Uni­ versity Senate Research Committee. For his visit to the Department ofMathematics and Statistics the authors are thankful to the Bowling Green State University.

In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. Fang, Kotz, and Ng presented a systematic study of multivariate elliptical distributions, however, they did not discuss the matrix variate case. Recently Fang and Zhang have summarized the results of generalized multivariate analysis which include vector as well as the matrix variate distributions. On the other hand, Fang and Anderson collected research papers on matrix variate elliptical distributions, many of them published for the first time in English. They published very rich material on the topic, but the results are given in paper form which does not provide a unified treatment of the theory. Therefore, it seemed appropriate to collect the most important results on the theory of matrix variate elliptically contoured distributions available in the literature and organize them in a unified manner that can serve as an introduction to the subject. The book will be useful for researchers, teachers, and graduate students in statistics and related fields whose interests involve multivariate statistical analysis. Parts of this book were presented by Arjun K Gupta as a one semester course at Bowling Green State University. Some new results have also been included which generalize the results in Fang and Zhang. Knowledge of matrix algebra and statistics at the level of Anderson is assumed. However, Chapter 1 summarizes some results of matrix algebra.

This revised and expanded second edition is an in-depth study of the change point problem from a general point of view, as well as a further examination of change point analysis of the most commonly used statistical models. Change point problems are encountered in such disciplines as economics, finance, medicine, psychology, signal processing, and geology, to mention only several. More recently, change point analysis has been found in extensive applications related to analyzing biomedical imaging data, array Comparative Genomic Hybridization (aCGH) data, and gene expression data. The exposition throughout the work is clear and systematic, with a great deal of introductory material included. Different models are presented in each chapter, including gamma and exponential models, rarely examined thus far in the literature. Extensive examples throughout the text emphasize key concepts and different methodologies used, namely the likelihood ratio criterion as well as the Bayesian and information criterion approaches. New examples of change point analysis in modern molecular biology and other fields such as finance and air traffic control are added in this second edition. Also included are two new chapters on change points in the hazard function and other practical change point models such as the epidemic change point model and a smooth-and-abrupt change point model. An up-to-date comprehensive bibliography and two indices round out the work.

Probability models are now a vital componentof every scienti c investigation. This book is intended to introduce basic ideas in stochastic modeling, with emphasis on models and techniques. These models lead to well-known parametric lifetime distributions, such as exponential, Weibull, and gamma distributions, as well as the change-point and mixture models. They also motivate us to consider more general notions of nonparametric lifetime distribution classes. Particular attention has been paid to their applications in reliability, insurance mathematics, and economics. The following topics are the focus in this volume: 1. Exponential Distributions and the Poisson Process; 2. Parametric Lifetime Distributions; 3. Nonparametric Lifetime Distribution Classes; 4. Multivariate Exponential Extensions; 5. Association and Dependence; 6. Renewal Theory; 7. Applications to Reliability, Insurance, Finance, and Credit Risk. Chapter1providesnotationandbasicresultsinprobabilitytheorythatareneeded in the consequent chapters. Chapters 2 and 3 are devoted to models related to exponential distribution and Poisson processes. Particular attentions is paid to the characterizations of exponential distribution and the Poisson process. Two of the most important properties that characterize exponential distribution: the lack of memory property and constant failure rate are discussed in detail. Then the g- eralizations of exponential distribution are examined in three directions: through its parametric form that leads to parametric families of lifetime distributions; via notionsof aging(such as monotonefailure rate) that lead to a varietyof lifetime d- tribution classes; and through lifetime distributions of multiple component systems that lead to multivariate (mainly bivariate) exponential extension.

Probability models are now a vital componentof every scienti c investigation. This book is intended to introduce basic ideas in stochastic modeling, with emphasis on models and techniques. These models lead to well-known parametric lifetime distributions, such as exponential, Weibull, and gamma distributions, as well as the change-point and mixture models. They also motivate us to consider more general notions of nonparametric lifetime distribution classes. Particular attention has been paid to their applications in reliability, insurance mathematics, and economics. The following topics are the focus in this volume: 1. Exponential Distributions and the Poisson Process; 2. Parametric Lifetime Distributions; 3. Nonparametric Lifetime Distribution Classes; 4. Multivariate Exponential Extensions; 5. Association and Dependence; 6. Renewal Theory; 7. Applications to Reliability, Insurance, Finance, and Credit Risk. Chapter1providesnotationandbasicresultsinprobabilitytheorythatareneeded in the consequent chapters. Chapters 2 and 3 are devoted to models related to exponential distribution and Poisson processes. Particular attentions is paid to the characterizations of exponential distribution and the Poisson process. Two of the most important properties that characterize exponential distribution: the lack of memory property and constant failure rate are discussed in detail. Then the g- eralizations of exponential distribution are examined in three directions: through its parametric form that leads to parametric families of lifetime distributions; via notionsof aging(such as monotonefailure rate) that lead to a varietyof lifetime d- tribution classes; and through lifetime distributions of multiple component systems that lead to multivariate (mainly bivariate) exponential extension.

A milestone in the published literature on the subject, this first-ever Handbook of Beta Distribution and Its Applications clearly enumerates the properties of beta distributions and related mathematical notions. It summarizes modern applications in a variety of fields, reviews up-and-coming progress from the front lines of statistical research and practice, and demonstrates the applicability of beta distributions in fields such as economics, quality control, soil science, and biomedicine. The book discusses the centrality of beta distributions in Bayesian inference, the beta-binomial model and applications of the beta-binomial distribution, and applications of Dirichlet integrals.

A milestone in the published literature on the subject, this first-ever Handbook of Beta Distribution and Its Applications clearly enumerates the properties of beta distributions and related mathematical notions. It summarizes modern applications in a variety of fields, reviews up-and-coming progress from the front lines of statistical research and

to Actuarial Mathematics by A. K. Gupta Bowling Green State University, Bowling Green, Ohio, U. S. A. and T. Varga National Pension Insurance Fund. Budapest, Hungary SPRINGER-SCIENCE+BUSINESS MEDIA, B. V. A C. I. P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5949-9 ISBN 978-94-017-0711-4 (eBook) DOI 10. 1007/978-94-017-0711-4 Printed on acid-free paper All Rights Reserved © 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. To Alka, Mita, and Nisha AKG To Terezia and Julianna TV TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTER 1. FINANCIAL MATHEMATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 2. Present Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1. 3. Annuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 CHAPTER 2. MORTALITy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2. 1Survival Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2. 2. Actuarial Functions of Mortality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2. 3. Mortality Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 CHAPTER 3. LIFE INSURANCES AND ANNUITIES . . . . . . . . . . . . . . . . . . . . . 112 3. 1. Stochastic Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3. 2. Pure Endowments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3. 3. Life Insurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3. 4. Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3. 5. Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 CHAPTER 4. PREMIUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4. 1. Net Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4. 2. Gross Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Vll CHAPTER 5. RESERVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5. 1. Net Premium Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5. 2. Mortality Profit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5. 3. Modified Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 ANSWERS TO ODD-NuMBERED PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Information-Statistical Data Mining: Warehouse Integration with Examples of Oracle Basics is written to introduce basic concepts, advanced research techniques, and practical solutions of data warehousing and data mining for hosting large data sets and EDA. This book is unique because it is one of the few in the forefront that attempts to bridge statistics and information theory through a concept of patterns. Information-Statistical Data Mining: Warehouse Integration with Examples of Oracle Basics is designed for a professional audience composed of researchers and practitioners in industry. This book is also suitable as a secondary text for graduate-level students in computer science and engineering.

Information-Statistical Data Mining: Warehouse Integration with Examples of Oracle Basics is written to introduce basic concepts, advanced research techniques, and practical solutions of data warehousing and data mining for hosting large data sets and EDA. This book is unique because it is one of the few in the forefront that attempts to bridge statistics and information theory through a concept of patterns. Information-Statistical Data Mining: Warehouse Integration with Examples of Oracle Basics is designed for a professional audience composed of researchers and practitioners in industry. This book is also suitable as a secondary text for graduate-level students in computer science and engineering.

Information-Statistical Data Mining: Warehouse Integration with Examples of Oracle Basics is written to introduce basic concepts, advanced research techniques, and practical solutions of data warehousing and data mining for hosting large data sets and EDA. This book is unique because it is one of the few in the forefront that attempts to bridge statistics and information theory through a concept of patterns. Information-Statistical Data Mining: Warehouse Integration with Examples of Oracle Basics is designed for a professional audience composed of researchers and practitioners in industry. This book is also suitable as a secondary text for graduate-level students in computer science and engineering.

Elliptically Contoured Models in Statistics and Portfolio Theory fully revises the first detailed introduction to the theory of matrix variate elliptically contoured distributions. There are two additional chapters, and all the original chapters of this classic text have been updated. Resources in this book will be valuable for researchers, practitioners, and graduate students in statistics and related fields of finance and engineering. Those interested in multivariate statistical analysis and its application to portfolio theory will find this text immediately useful. ​In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Elliptical distributions have also increased their popularity in finance because of the ability to model heavy tails usually observed in real data. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. A noteworthy function of this book is the collection of the most important results on the theory of matrix variate elliptically contoured distributions that were previously only available in the journal-based literature. The content is organized in a unified manner that can serve an a valuable introduction to the subject.​

Elliptically Contoured Models in Statistics and Portfolio Theory fully revises the first detailed introduction to the theory of matrix variate elliptically contoured distributions. There are two additional chapters, and all the original chapters of this classic text have been updated. Resources in this book will be valuable for researchers, practitioners, and graduate students in statistics and related fields of finance and engineering. Those interested in multivariate statistical analysis and its application to portfolio theory will find this text immediately useful. ​In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Elliptical distributions have also increased their popularity in finance because of the ability to model heavy tails usually observed in real data. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. A noteworthy function of this book is the collection of the most important results on the theory of matrix variate elliptically contoured distributions that were previously only available in the journal-based literature. The content is organized in a unified manner that can serve an a valuable introduction to the subject.

​

A milestone in the published literature on the subject, this first-ever Handbook of Beta Distribution and Its Applications clearly enumerates the properties of beta distributions and related mathematical notions. It summarizes modern applications in a variety of fields, reviews up-and-coming progress from the front lines of statistical research and

Elliptically Contoured Models in Statistics and Portfolio Theory fully revises the first detailed introduction to the theory of matrix variate elliptically contoured distributions. There are two additional chapters, and all the original chapters of this classic text have been updated. Resources in this book will be valuable for researchers, practitioners, and graduate students in statistics and related fields of finance and engineering. Those interested in multivariate statistical analysis and its application to portfolio theory will find this text immediately useful. ​In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Elliptical distributions have also increased their popularity in finance because of the ability to model heavy tails usually observed in real data. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. A noteworthy function of this book is the collection of the most important results on the theory of matrix variate elliptically contoured distributions that were previously only available in the journal-based literature. The content is organized in a unified manner that can serve an a valuable introduction to the subject.​

The death of Professor K. C. Sreedharan Pillai on June 5, 1985 was a heavy loss to many statisticians all around the world. This volume is dedicated to his memory in recog- nition of his many contributions in multivariate statis- tical analysis. It brings together eminent statisticians Working in multivariate analysis from around the world. The research and expository papers cover a cross-section of recent developments in the field. This volume is especially useful to researchers and to those who want to keep abreast of the latest directions in multivariate statistical analysis. I am grateful to the authors from so many different countries and research institutions who contributed to this volume. I wish to express my appreciation to all those who have reviewed the papers. The list of people include Professors T. C. Chang, So-Hsiang Chou, Dipak K. Dey, Peter Hall, Yu-Sheng Hsu, J. D. Knoke, W. J. Krzanowski, Edsel Pena, Bimal K. Sinha, Dennis L. Young, Drs. K. Krishnamoorthy, D. K. Nagar, and Messrs. Alphonse Amey, Chi-Chin Chao and Samuel Ofori-Nyarko. I wish to thank Professors Shanti S. Gupta and James 0. Berger for their keen interest and encouragement.Thanks are also due to Cynthia Patterson for her help and Reidel Publishing Com~any for their cooperation in bringing this volume out.

This volume contains the Proceedings of the Advanced Symposium on Multivariate Modeling and Data Analysis held at the 64th Annual Heeting of the Virginia Academy of Sciences (VAS)--American Statistical Association's Vir­ ginia Chapter at James Madison University in Harrisonburg. Virginia during Hay 15-16. 1986. This symposium was sponsored by financial support from the Center for Advanced Studies at the University of Virginia to promote new and modern information-theoretic statist­ ical modeling procedures and to blend these new techniques within the classical theory. Multivariate statistical analysis has come a long way and currently it is in an evolutionary stage in the era of high-speed computation and computer technology. The Advanced Symposium was the first to address the new innovative approaches in multi­ variate analysis to develop modern analytical and yet practical procedures to meet the needs of researchers and the societal need of statistics. vii viii PREFACE Papers presented at the Symposium by e1l11lJinent researchers in the field were geared not Just for specialists in statistics, but an attempt has been made to achieve a well balanced and uniform coverage of different areas in multi­ variate modeling and data analysis. The areas covered included topics in the analysis of repeated measurements, cluster analysis, discriminant analysis, canonical cor­ relations, distribution theory and testing, bivariate densi ty estimation, factor analysis, principle component analysis, multidimensional scaling, multivariate linear models, nonparametric regression, etc.

to Actuarial Mathematics by A. K. Gupta Bowling Green State University, Bowling Green, Ohio, U. S. A. and T. Varga National Pension Insurance Fund. Budapest, Hungary SPRINGER-SCIENCE+BUSINESS MEDIA, B. V. A C. I. P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5949-9 ISBN 978-94-017-0711-4 (eBook) DOI 10. 1007/978-94-017-0711-4 Printed on acid-free paper All Rights Reserved © 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. To Alka, Mita, and Nisha AKG To Terezia and Julianna TV TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTER 1. FINANCIAL MATHEMATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 2. Present Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1. 3. Annuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 CHAPTER 2. MORTALITy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2. 1Survival Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2. 2. Actuarial Functions of Mortality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2. 3. Mortality Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 CHAPTER 3. LIFE INSURANCES AND ANNUITIES . . . . . . . . . . . . . . . . . . . . . 112 3. 1. Stochastic Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3. 2. Pure Endowments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3. 3. Life Insurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3. 4. Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3. 5. Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 CHAPTER 4. PREMIUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4. 1. Net Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4. 2. Gross Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Vll CHAPTER 5. RESERVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5. 1. Net Premium Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5. 2. Mortality Profit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5. 3. Modified Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 ANSWERS TO ODD-NuMBERED PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The death of Professor K. C. Sreedharan Pillai on June 5, 1985 was a heavy loss to many statisticians all around the world. This volume is dedicated to his memory in recog- nition of his many contributions in multivariate statis- tical analysis. It brings together eminent statisticians Working in multivariate analysis from around the world. The research and expository papers cover a cross-section of recent developments in the field. This volume is especially useful to researchers and to those who want to keep abreast of the latest directions in multivariate statistical analysis. I am grateful to the authors from so many different countries and research institutions who contributed to this volume. I wish to express my appreciation to all those who have reviewed the papers. The list of people include Professors T. C. Chang, So-Hsiang Chou, Dipak K. Dey, Peter Hall, Yu-Sheng Hsu, J. D. Knoke, W. J. Krzanowski, Edsel Pena, Bimal K. Sinha, Dennis L. Young, Drs. K. Krishnamoorthy, D. K. Nagar, and Messrs. Alphonse Amey, Chi-Chin Chao and Samuel Ofori-Nyarko. I wish to thank Professors Shanti S. Gupta and James 0. Berger for their keen interest and encouragement.Thanks are also due to Cynthia Patterson for her help and Reidel Publishing Com~any for their cooperation in bringing this volume out.

This volume contains the Proceedings of the Advanced Symposium on Multivariate Modeling and Data Analysis held at the 64th Annual Heeting of the Virginia Academy of Sciences (VAS)--American Statistical Association''s Vir­ ginia Chapter at James Madison University in Harrisonburg. Virginia during Hay 15-16. 1986. This symposium was sponsored by financial support from the Center for Advanced Studies at the University of Virginia to promote new and modern information-theoretic statist­ ical modeling procedures and to blend these new techniques within the classical theory. Multivariate statistical analysis has come a long way and currently it is in an evolutionary stage in the era of high-speed computation and computer technology. The Advanced Symposium was the first to address the new innovative approaches in multi­ variate analysis to develop modern analytical and yet practical procedures to meet the needs of researchers and the societal need of statistics. vii viii PREFACE Papers presented at the Symposium by e1l11lJinent researchers in the field were geared not Just for specialists in statistics, but an attempt has been made to achieve a well balanced and uniform coverage of different areas in multi­ variate modeling and data analysis. The areas covered included topics in the analysis of repeated measurements, cluster analysis, discriminant analysis, canonical cor­ relations, distribution theory and testing, bivariate densi ty estimation, factor analysis, principle component analysis, multidimensional scaling, multivariate linear models, nonparametric regression, etc.

In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. Fang, Kotz, and Ng presented a systematic study of multivariate elliptical distributions, however, they did not discuss the matrix variate case. Recently Fang and Zhang have summarized the results of generalized multivariate analysis which include vector as well as the matrix variate distributions. On the other hand, Fang and Anderson collected research papers on matrix variate elliptical distributions, many of them published for the first time in English. They published very rich material on the topic, but the results are given in paper form which does not provide a unified treatment of the theory. Therefore, it seemed appropriate to collect the most important results on the theory of matrix variate elliptically contoured distributions available in the literature and organize them in a unified manner that can serve as an introduction to the subject. The book will be useful for researchers, teachers, and graduate students in statistics and related fields whose interests involve multivariate statistical analysis. Parts of this book were presented by Arjun K Gupta as a one semester course at Bowling Green State University. Some new results have also been included which generalize the results in Fang and Zhang. Knowledge of matrix algebra and statistics at the level of Anderson is assumed. However, Chapter 1 summarizes some results of matrix algebra.

This volume contains the Proceedings of the Advanced Symposium on Multivariate Modeling and Data Analysis held at the 64th Annual Heeting of the Virginia Academy of Sciences (VAS)--American Statistical Association's Vir­ ginia Chapter at James Madison University in Harrisonburg. Virginia during Hay 15-16. 1986. This symposium was sponsored by financial support from the Center for Advanced Studies at the University of Virginia to promote new and modern information-theoretic statist­ ical modeling procedures and to blend these new techniques within the classical theory. Multivariate statistical analysis has come a long way and currently it is in an evolutionary stage in the era of high-speed computation and computer technology. The Advanced Symposium was the first to address the new innovative approaches in multi­ variate analysis to develop modern analytical and yet practical procedures to meet the needs of researchers and the societal need of statistics. vii viii PREFACE Papers presented at the Symposium by e1l11lJinent researchers in the field were geared not Just for specialists in statistics, but an attempt has been made to achieve a well balanced and uniform coverage of different areas in multi­ variate modeling and data analysis. The areas covered included topics in the analysis of repeated measurements, cluster analysis, discriminant analysis, canonical cor­ relations, distribution theory and testing, bivariate densi ty estimation, factor analysis, principle component analysis, multidimensional scaling, multivariate linear models, nonparametric regression, etc.

In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. Fang, Kotz, and Ng presented a systematic study of multivariate elliptical distributions, however, they did not discuss the matrix variate case. Recently Fang and Zhang have summarized the results of generalized multivariate analysis which include vector as well as the matrix variate distributions. On the other hand, Fang and Anderson collected research papers on matrix variate elliptical distributions, many of them published for the first time in English. They published very rich material on the topic, but the results are given in paper form which does not provide a unified treatment of the theory. Therefore, it seemed appropriate to collect the most important results on the theory of matrix variate elliptically contoured distributions available in the literature and organize them in a unified manner that can serve as an introduction to the subject. The book will be useful for researchers, teachers, and graduate students in statistics and related fields whose interests involve multivariate statistical analysis. Parts of this book were presented by Arjun K Gupta as a one semester course at Bowling Green State University. Some new results have also been included which generalize the results in Fang and Zhang. Knowledge of matrix algebra and statistics at the level of Anderson is assumed. However, Chapter 1 summarizes some results of matrix algebra.