About the Authors Prem C. Consul is professor emeritus in the Department of Mathematics and Statistics at the University of Calgary in Calgary, Alberta, Canada. Dr. Consul received his Ph.D. degree in mathematical statistics and a master’s degree in mathematics from Dr. Bhimrao Ambedkar University (formerly Agra University), Agra in India. Prior to joining the University of Calgary in 1967, he was a professor at the University of Libya for six years, the principal of a degree college in India for four years, and was a professor and assistant professor in degree colleges for ten years. He was on the editorial board of Communications in Statistics for many years. He is a well-known researcher in distribution theory and multivariate analysis. He published three books prior to the present one. Felix Famoye is a professor and a consulting statistician in the Department of Mathematics at Central Michigan University in Mount Pleasant, Michigan, USA. Prior to joining the staff at Central Michigan University, he was a postdoctoral research fellow and an associate at the University of Calgary, Calgary, Canada. He received his B.SC. (Honors) degree in statistics from the University of Ibadan, Ibadan, Nigeria. He received his Ph.D. degree in statistics from the University of Calgary under the Canadian Commonwealth Scholarship. Dr. Famoye is a well-known researcher in statistics. He has been a visiting scholar at the University of Kentucky, Lexington and University of North Texas Health Science Center, Fort Worth, Texas. Prem C. Consul Felix Famoye Lagrangian Probability Distributions Birkhauser¨ Boston • Basel • Berlin Prem C. Consul Felix Famoye University of Calgary Central Michigal University Department of Mathematics and Statistics Department of Mathematics Calgary, Alberta T2N 1N4 Mount Pleasant, MI 48859 Canada USA Cover design by Alex Gerasev. Mathematics Subject Classification (2000): 60Exx, 60Gxx, 62Exx, 62Pxx (primary); 60E05, 60G50, 60J75, 60J80, 60K25, 62E10, 62P05, 62P10, 62P12, 62P30, 65C10 (secondary) Library of Congress Control Number: 2005tba ISBN-10 0-8176-4365-6 eISBN 0-8176- ISBN-13 978-0-8176-4365-2 Printed on acid-free paper. c 2006 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA), 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. Printed in the United States of America. (KeS/MP) 987654321 www.birkhauser.com Joseph Louis Lagrange (1736–1813) Joseph Louis Lagrange was one of the two great mathematicians of the eighteenth century. He was born in France and was appointed professor at the age of 19. He helped in founding the Royal Academy of Sciences at the Royal Artillery School in 1757. He was very close to the famous mathematician Euler, who appreciated his work immensely. When Euler left the Berlin Academy of Science in 1766, Lagrange succeeded him as director of mathematics. He left Berlin in 1787 and became a member of the Paris Academy of Science and remained there for the rest of his career. He helped in the establishment of Ecole´ Polytechnique and taught there for some time. He survived the French revolution, and Napoleon appointed him to the Legion of Honour and Count of the Empire. Lagrange had given two formulae for the expansion of the function f (z) in a power series of u when z = ug(z) (memoires´ de l’Acad. Roy. des Sci. Berlin, 24, 1768, 251) which have been extensively used by various researchers for developing the class of Lagrangian probability models and its families described in this book. Lagrange developed the calculus of variations, which was very effective in dealing with mechanics. His work Mecanique´ Analytique (1788), summarizing all the work done earlier in mechanics, contained unique methods using differential equations and mathematical analysis. He created Lagrangian mechanics, provided many new solutions and theorems in number the- ory, the method of Lagrangian multipliers, and numerous other results which were found to be extremely useful. To Shakuntla Consul and Busola Famoye Foreword It is indeed an honor and pleasure to write a Foreword to the book on Lagrangian Probability Distributions by P. C. Consul and Felix Famoye. This book has been in the making for some time and its appearance marks an important milestone in the series of monographs on basic statistical distributions which have originated in the second half of the last century. The main impetus for the development of an orderly investigation of statistical distributions and their applications was the International Symposium on Classical and Contagious Discrete Distributions, organized by G. P. Patil in August of 1969–some forty years ago—with the active participation of Jerzy Neyman and a number of other distinguished researchers in the field of statistical distributions. This was followed by a number of international conferences on this topic in various lo- cations, including Trieste, Italy and Calgary, Canada. These meetings, which took place dur- ing the period of intensive development of computer technology and its rapid penetration into statistical analyses of numerical data, served inter alia, as a shield, resisting the growing at- titude and belief among theoreticians and practitioners that it may perhaps be appropriate to de-emphasize the parametric approach to statistical models and concentrate on less invasive nonparametric methodology. However, experience has shown that parametric models cannot be ignored, in particular in problems involving a large number of variables, and that without a distributional “saddle,” the ride towards revealing and analyzing the structure of data repre- senting a certain “real world” phenomenon turns out to be burdensome and often less reliable. P. C. Consul and his former student and associate for the last 20 years Felix Famoye were at the forefront of intensive study of statistical distribution—notably the discrete one—during the golden era of the last three decades of the twentieth century. In addition to numerous papers, both of single authorship and jointly with leading scholars, P. C. Consul opened new frontiers in the field of statistical distributions and applications by discovering many useful and elegant distributions and simultaneously paying attention to com- putational aspects by developing efficient and relevant computer programs. His earlier (1989) 300-page volume on Generalized Poisson Distributions exhibited very substantial erudition and the ability to unify and coordinate seemingly isolated results into a coherent and reader-friendly text (in spite of nontrivial and demanding concepts and calculations). The comprehensive volume under consideration, consisting of 16 chapters, provides a broad panorama of Lagrangian probability distributions, which utilize the series expansion of an an- alytic function introduced by the well-known French mathematician J. L. Lagrange (1736– 1813) in 1770 and substantially extended by the German mathematician H. B¨urmann in 1779. viii Foreword A multivariate extension was developed by I. J. Good in 1955, but the definition and basic properties of the Lagrangian distributions are due to P. C. Consul, who in collaboration with R. L. Shenton wrote in the early 1970s in a number of pioneering papers with detailed discus- sion of these distributions. This book is a welcome addition to the literature on discrete univariate and multivariate distributions and is an important source of information on numerous topics associated with powerful new tools and probabilistic models. The wealth of materials is overwhelming and the well-organized, lucid presentation is highly commendable. Our thanks go to the authors for their labor of love, which will serve for many years as a textbook, as well as an up-to-date handbook of the results scattered in the periodical literature and as an inspiration for further research in an only partially explored field. Samuel Kotz The George Washington University, U.S.A. December 1, 2003 Preface Lagrange had given two expansions for a function towards the end of the eighteenth century, but they were used very little. Good (1955, 1960, 1965) did the pioneering work by developing their multivariate generalization and by applying them effectively to solve a number of impor- tant problems. However, his work did not generate much interest among researchers, possibly because the problems he considered were complex and his presentation was too concise. The Lagrange expansions can be used to obtain very useful numerous probability models. During the last thirty years, a very large number of research papers has been published by numerous researchers in various journals on the class of Lagrangian probability distributions, its interesting families, and related models. These probability models have been applied to many real life situations including, but not limited to, branching processes, queuing processes, stochastic processes, environmental toxicology, diffusion of information, ecology, strikes in industries, sales of new products, and amounts of production for optimum profits. The first author of this book was the person who defined the Lagrangian probability distrib- utions and who had actively started research on some of these models, in collaboration with his associates, about thirty years ago. After the appearance of six research papers in quick succes- sion until 1974, other researchers were anxious to know more. He vividly remembers the day in the 1974 NATO Conference at Calgary, when a special meeting was held and the first au- thor was asked for further elucidation of the work on Lagrangian probability models.