Review of Halphen Distribution Family with Application in Hydrological Frequency Analysis by Salahedddine El Adlouni and Bernard Bobée

In his foreword, Professor Jacques Bernier provides a glimpse of the history of the Halphen distribution family originated by Etienne Halphen who worked at Societe Hydrotechnique de France (SHF) in France from 1940 to 1954. In 1940 SHF, headed by Pierre Masse, was created for developing probabilistic and statistical techniques for the management of hydroelectric resources. Inspired by Professor Bernier, Professors El Adlouni and Bobee took on the challenge of writing this monograph on the Halphen distribution family that comprises three three-parameter distributions. Although these distributions possess a number of desiderata, they have been less than popular in the hydrologic literature, let alone in practice. Some of their desirable characteristics are generality, flexibility, versatility, and sufficiency of estimators. Both authors have extensively worked on the Halphen distributions and are therefore uniquely qualified to write this much needed book.

The subject matter of the book spans six chapters and three appendices. Chapter one provides a short history of frequency analysis using Halphen distributions. Three Halphen distributions, called HA, HB, and HIB, are presented in chapters two to four. Detailed in these chapters are mathematical and statistical properties, asymptotic behavior and model selection, and parameter estimation methods for these distributions. Providing a short historical context, chapter two derives mode, moments, and moment ratios, and discusses sufficient statistics and special cases of the three distributions. The chapter is then summarized with key points detailed in the chapter. On the whole, the chapter is written with rigor and clarity.

Chapter three deals with the asymptotic behavior of the Halphen distributions. Starting with classes of heavy-tailed distributions, it goes on to discuss the asymptotic behavior and decision support systems, comparing the Halphen family with the generalized extreme value system, and testing the HA and HB distributions against the gamma distribution. The chapter is written in a lucid and easy to understand manner.

Parameter estimation methods for the Halphen distributions constitute the subject matter of chapter four. The maximum likelihood method for estimating parameters of the HA, HB, and HIB distributions is discussed first. Then, the method of moments is presented, which is followed by the iterative mixed method. The Bayesian approach for the Halphen distribution is presented in the concluding part of the chapter. It is a well-written chapter.

Monte Carlo simulation for the Halphen distributions is treated in chapter five. Introducing problems related to statistical inference with these distributions, the chapter goes on to discuss the random variable generation, acceptance-rejection approach for the Halphen distributions, and the Markov Chain Monte Carlo method using the Metropolis–Hastings algorithm. Overall, the chapter is insightful and valuable.

Chapter six discusses MATLAB codes for Halphen fitting and generating, illustrated with a case study. This chapter demonstrates the practical application and the value of the Halphen family. Appendix A deals with Bessel functions, Appendix B with exponential factorial function, and Appendix C with decision support system.

This short monograph is beautifully written and is insightful. It contains much useful material which is detailed with rigor, clarity, and easy comprehension. The book will be highly useful for statistical hydrologists. Professors El Adlouni and Bobee deserve significant applause and congratulations for writing this monograph. It is hoped that this monograph will help find a notable place in the hydrologic literature and practice that the Halphen distribution family deserves.