Statistical Size Distributions in Economics and Actuarial Sciences WILEY SERIES IN PROBABILITYAND STATISTICS Established by WALTER A. SHEWHARTand SAMUEL S. WILKS Editors: David J. Balding, Peter Bloomfield, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, Louise M. Ryan, David W. Scott, Adrian F. M. Smith, Jozef L. Teugels; Editors Emeriti: Vic Barnett, J. Stuart Hunter, David G. Kendall A complete list of the titles in this series appears at the end of this volume. Statistical Size Distributions in Economics and Actuarial Sciences CHRISTIAN KLEIBER Universita¨t Dortmund, Germany SAMUEL KOTZ The George Washington University A JOHN WILEYAND SONS, INC., PUBLICATION Copyrightq2003byJohnWiley&Sons,Inc.Allrightsreserved. PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey. PublishedsimultaneouslyinCanada. 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LibraryofCongressCataloging-in-PublicationData: Kleiber,Christian,1966- Statisticalsizedistributionsineconomicsandactuarialsciences/ChristianKleiber,SamuelKotz. p.cm.—(Wileyseriesinprobabilityandstatistics) Includesbibliographicalreferencesandindex. ISBN0-471-15064-9(cloth) 1.Distribution(Economictheory)3.Economics,Mathematical. 4.Insurance–Mathematics.I.Kotz,Samuel.II.Title.III.Series. HB523.K552003 339.202–dc21 2003041140 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 Contents Preface ix Acknowledgments xi 1 Introduction 1 1.1 Our Aims, 1 1.2 Types of Economic Size Distributions, 5 1.3 Brief History of the Models for Studying Economic Size Distributions, 6 1.4 Stochastic Process Models for Size Distributions, 14 2 General Principles 19 2.1 Some Concepts from Economics, 19 2.2 Hazard Rates, Mean Excess Functions, and Tailweight, 43 2.3 Systems of Distributions, 49 2.4 Generating Systems of Income Distributions, 55 3 Pareto Distributions 59 3.1 Definition, 59 3.2 History and Genesis, 61 3.3 Moments and Other Basic Properties, 70 3.4 Characterizations, 74 3.5 Lorenz Curve and Inequality Measures, 77 3.6 Estimation, 79 3.7 Empirical Results, 91 3.8 Stoppa Distributions, 94 v CONTENTS vi 3.9 Conic Distribution, 97 3.10 A “Log-Adjusted” Pareto distribution, 99 3.11 Stable Distributions, 101 3.12 Further Pareto-Type Distributions, 103 4 Lognormal Distributions 107 4.1 Definition, 107 4.2 History and Genesis, 108 4.3 Moments and Other Basic Properties, 110 4.4 Characterizations, 115 4.5 Lorenz Curve and Inequality Measures, 115 4.6 Estimation, 118 4.7 Three- and Four-Parameter Lognormal Distributions, 121 4.8 Multivariate Lognormal Distribution, 124 4.9 Empirical Results, 126 4.10 Generalized Lognormal Distribution, 131 4.11 An Asymmetric Log-Laplace Distribution, 140 4.12 Related Distributions, 143 5 Gamma-Type Size Distributions 147 5.1 Generalized Gamma Distribution, 147 5.2 Gamma Distribution, 160 5.3 Log-Gamma Distribution, 168 5.4 Inverse Gamma (Vinci) Distribution, 172 5.5 Weibull Distribution, 174 5.6 Log-Gompertz Distribution, 179 6 Beta-Type Size Distributions 183 6.1 (Generalized) Beta Distribution of the Second Kind, 183 6.2 Singh–Maddala Distribution, 197 6.3 Dagum Distributions, 212 6.4 Fisk (Log-Logistic) and Lomax Distributions, 222 6.5 (Generalized) Beta Distribution of the First Kind, 230 7 Miscellaneous Size Distributions 235 7.1 Benini Distribution, 235 7.2 Davis Distribution, 238 7.3 Champernowne Distribution, 240 7.4 Benktander Distributions, 247 vii CONTENTS Appendix A Biographies 251 A.1 Vilfredo Federico Domaso Pareto, Marchese di Parigi, 252 A.2 Rodolfo Benini, 261 A.3 Max Otto Lorenz, 263 A.4 Corrado Gini, 265 A.5 Luigi Amoroso, 267 A.6 Raffaele D’Addario, 269 A.7 Robert Pierre Louis Gibrat, 271 A.8 David Gawen Champernowne, 273 Appendix B Data on Size Distributions 277 Appendix C Size Distributions 283 List of Symbols 287 References 289 Author Index 319 Subject Index 327 Preface This is a book about money, but it will not help you very much in learning how to make money. Rather, it will instruct you about the distribution of various kinds of income and their related economic size distributions. Specifically, we have painstakingly traced the numerous statistical models of income distribution, from the late nineteenth century when Vilfredo Pareto developed a bold and astonishing modelforthedistributionofpersonalincomeuntilthelatestmodelsdevelopedsome 100 years later. Our goal was to review, compare, and somehow connect all these models and to pinpoint the unfortunate lack of coordination among various researchers,whichhasresultedintheduplicationofeffortandwasteoftalentandto some extent has reduced the value of their contributions. We also discuss the size distributions of loss in actuarial applications that involve a number of distributions used for income purposes. An impatient reader may wish to consult the list of distributionscoveredinthisbookandtheirbasicpropertiespresentedinAppendixC. The task of compiling this interdisciplinary book took longer and was more arduous than originally anticipated. We have tried to describe the distributions outlined herewithin the contextof the personalities of their originatorssince in our opinionthepersonality,temperament,andbackgroundoftheauthorsciteddidaffect to some extent the nature and scope of their discoveries and contributions. We hope that our readers come to regard this book as a reliable source of information and we gladly welcome all efforts to bring any remaining errors to our attention. CHRISTIAN KLEIBER Dortmund,Germany SAMUEL KOTZ Washington,D.C. ix