Statistical Size Distributions in Economics and Actuarial Sciences
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More About This Title Statistical Size Distributions in Economics and Actuarial Sciences


A comprehensive account of economic size distributions around the world and throughout the years

In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions.

Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include:

  • Pareto distributions
  • Lognormal distributions
  • Gamma-type size distributions
  • Beta-type size distributions
  • Miscellaneous size distributions

Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.


CHRISTIAN KLEIBER, PhD, is assistant professor in the Department of Statistics at the University of Dortmund in Germany.

SAMUEL KOTZ, PhD, honorary Doctor of Science, is professor and research scholar at the Department of Engineering Management and Systems Engineering at George Washington University in Washington, D.C.




1. Introduction.

1.1 Our Aims.

1.2 Types of Economic Size Distributions.

1.3 A Brief Historical Sketch of the Models for Studying Economic Size Distributions.

1.4 Stochastic Process Models for Size Distributions.

2. General Principles.

2.1 Some Concepts from Economics.

2.2 Hazard Rates, Mean Excess Functions, and Tailweight.

2.3 Systems of Distributions.

2.4 Generating Systems of Income Distributions.

3. Pareto Distributions.

3.1 Definitions.

3.2 History and Genesis.

3.3 Moments and Other Basic Properties.

3.4 Characterizations.

3.5 Lorenz Curve and Inequality Measures.

3.6 Estimation.

3.7 Empirical Results.

3.8 Stoppa Distributions.

3.9 Conic Distribution.

3.10 A "log-adjusted" Pareto Distribution.

3.11 Stable Distributions.

3.12 Further Pareto-type Distributions.

4. Lognormal Distribution.

4.1 Definition.

4.2 History and Genesis.

4.3 Moments and Other Basic Properties.

4.4 Characterizations.

4.5 Lorenz Curve and Inequality Measures.

4.6 Estimation.

4.7 Three-and four-parameter Lognormal Distributions.

4.8 Multivariate Lognormal Distribution.

4.9 Empirical Results.

4.10 Generalized Lognormal Distribution.

4.11 An Asymmetric Log-Laplace Distribution.

4.12 Related Distributions.

5. Gamma-type Size Distributions.

5.1 Generalized Gamma Distribution.

5.2 Gamma Distribution.

5.3 Log-gamma Distribution.

5.4 Inverse Gamma (Vinci) Distribution.

5.5 Weibull Distribution.

5.6 Log-Gompertz Distribution.

6. Beta-type Size Distributions.

6.1 (Generalized) Beta Distribution of the Second Kind.

6.2 Singh-Maddala Distribution.

6.3 Dagum Distribution.

6.4 Fisk (log-logistic) and Lomax Distributions.

6.5 (Generalized) Beta Distribution of the First Kind.

7. Miscellaneous Size Distributions.

7.1 Benini Distribution.

7.2 Davis Distribution.

7.3 Champernowne Distribution.

7.4 Benktander Distributions.

Appendix A. Biographies.

A.1 Vilfredo Federico Domaso Pareto, Marchese di Parigi.

A.2 Rodolfo Benini.

A.3 Max Otto Lorenz.

A.4 Corrado Gini.

A.5 Luigi Amoroso.

A.6 Raffaele D’Addario.

A.7 Robert Pierre Louis Gibrat.

A.8 David Gawen Champernowne.

Appendix B. Data on Size Distributions.

Appendix C. Size Distributions.

List of Symbols.


Author Index.

Subject Index.


"Researchers and teachers in econometrics and actuarial statistics will find this book an interesting source of information." (Journal of the American Statistical Association, December 2004)

"The overall quality of the book is excellent. The material is well-written and well laid out…an indispensable, authoritative source of information…" (Zentralblatt MATH Database)

“…an indispensable, authoritative source of information on a remarkably diverse variety of parametric models for applied statisticians and researchers…” (Zentralblatt Math, Vol.1044, No.19, 2004)

"As a reference book, it is exceptionally well done, as are all of the projects undertaken by Prof. Kotz…The statistics profession has been greatly enriched by this particular effort." (Technometrics, November 2004)

"…the authors have gone a long way towards achieving their ideal that 'a useful book on this subject matter should be interesting.” (Short Book Reviews, August 2004)

“...traces the numerous statistical models of income distribution for Pareto’s model in the late 19th century to the latest ones.” (Quarterly of Applied Mathematics, Vol. LXII, No. 1, March 2004)