##
**The inverse Gaussian distribution. Statistical theory and applications.**
*(English)*
Zbl 0942.62011

Lecture Notes in Statistics. 137. New York, NY: Springer. xii, 347 p. (1999).

The Inverse Gaussian (IG) distribution is the primary issue of many statistical books, after the publication of the pioneering book of R. S. Chhikara and J. L. Folks, The inverse Gaussian distribution: Theory, methodology and applications. (1989; Zbl 0701.62009). This book about this distribution is the second book that this author has written about the IG distribution and aims at covering the statistical issues and data analysis, which were somewhat ignored in his previous monograph, The inverse Gaussian distribution: A case study in exponential families. (1993). Moreover, as the author points out his book aspires to update the material of the book by Chhikara and Folks. The book is divided into two parts. The first part covers all the essential theory, providing likewise certain illustrative examples and the second is devoted to applications of the IG distribution in several disciplines.

The first part consists of 7 chapters. The first chapter refers to the principal developments on the distribution from the beginning of the \(20^{\text{th}}\) century, defines the IG distribution, and cites some of its alternative versions, that may arise using different parametrizations proposed for its implementation in different scientific fields. The shape of the distribution for different combinations of the parameters is investigated, too. Moreover, this chapter examines the limiting behavior of the distribution, deals with the sampling distributions that are associated with it, considers conditional distributions and concludes with some Bayesian aspects involving the assessment of the joint and the marginal posterior distributions of the parameters, for different choices of the prior distribution.

The second chapter addresses the issue of the estimation of the two parameters of the IG distribution. Uniformly minimum variance unbiased estimators of the parameters and some functions of them are provided. Moreover, the issue of estimation of the parameters of a more general form of the IG distribution, which involves a third parameter, is considered. Finally, it deals with the estimation of the (two-parameter) IG distribution in cases where the extreme observations from both sides are truncated.

The third chapter discusses some significance tests about the two parameters of the distribution. It commences with likelihood ratio tests for each of the two parameters separately, distinguishing between the cases where the other parameter is known or unknown. Furthermore, it deals with tests for the drift of a Brownian motion process, with two sample tests – again separately for the cases where the parameters are known or unknown, and with the construction of confidence intervals for the true value of the parameters.

The fourth chapter is devoted to some sequential methods. The merit of such methods is that on average they lead to sample sizes smaller than those required for implementing the standard techniques.

The fifth chapter is about reliability and survival analysis. It discusses methods of point estimation of the reliability function (based on classical or Bayesian approaches), the construction of confidence intervals and tolerance limits. Finally, it deals with point and interval estimation of the hazard rate and the critical time.

The sixth chapter, much shorter in comparison to the other chapters, outlines some goodness of fit procedures. In particular, the modifications of certain prominent tests such as Kolmogorov-Smirnov, Anderson-Darling, CramĂ©r Von Mises and Watson test, for the IG case are discussed. It has to be remarked that this chapter ignores graphical techniques useful for distribution fitting, as \(Q-Q\) plots.

The seventh chapter is the last chapter of the first part. It is the most extensive, and is connected with compound distributions, which may arise, as a result of mixing the IG distribution with some other broadly used distributions such as the Poisson, the normal and the exponential distributions. The resultant compound distributions are very useful whenever one needs to fit a distribution to long-tailed data. In addition, there is a description of the implementation of the mixture of Poisson and IG distributions as an error distribution of a linear regression model, which is appropriate for modeling overdispersed count data.

The second part includes a plethora of applications in several disciplines as diverse as life testing, regression analysis, physiology, entomology, control charts, demography.

The first part of the book, which as previously mentioned, covers theoretical issues, contains a large number of theorems whose proofs require of the reader a strong background on distributions, and statistical theory. Otherwise, its reading becomes hard, in as much as it does not comprise an introductory chapter devoted to the essential theory. On the other hand, the comprehension of the second part of the book is relatively easier. In addition, the second part can be beneficial to many scientists who would wish to utilize the IG distribution in their field of research.

Finally, it has to be noted that the book contains a vast bibliography related to the IG distribution, and can thus serve as a valuable tool for the researcher.

The first part consists of 7 chapters. The first chapter refers to the principal developments on the distribution from the beginning of the \(20^{\text{th}}\) century, defines the IG distribution, and cites some of its alternative versions, that may arise using different parametrizations proposed for its implementation in different scientific fields. The shape of the distribution for different combinations of the parameters is investigated, too. Moreover, this chapter examines the limiting behavior of the distribution, deals with the sampling distributions that are associated with it, considers conditional distributions and concludes with some Bayesian aspects involving the assessment of the joint and the marginal posterior distributions of the parameters, for different choices of the prior distribution.

The second chapter addresses the issue of the estimation of the two parameters of the IG distribution. Uniformly minimum variance unbiased estimators of the parameters and some functions of them are provided. Moreover, the issue of estimation of the parameters of a more general form of the IG distribution, which involves a third parameter, is considered. Finally, it deals with the estimation of the (two-parameter) IG distribution in cases where the extreme observations from both sides are truncated.

The third chapter discusses some significance tests about the two parameters of the distribution. It commences with likelihood ratio tests for each of the two parameters separately, distinguishing between the cases where the other parameter is known or unknown. Furthermore, it deals with tests for the drift of a Brownian motion process, with two sample tests – again separately for the cases where the parameters are known or unknown, and with the construction of confidence intervals for the true value of the parameters.

The fourth chapter is devoted to some sequential methods. The merit of such methods is that on average they lead to sample sizes smaller than those required for implementing the standard techniques.

The fifth chapter is about reliability and survival analysis. It discusses methods of point estimation of the reliability function (based on classical or Bayesian approaches), the construction of confidence intervals and tolerance limits. Finally, it deals with point and interval estimation of the hazard rate and the critical time.

The sixth chapter, much shorter in comparison to the other chapters, outlines some goodness of fit procedures. In particular, the modifications of certain prominent tests such as Kolmogorov-Smirnov, Anderson-Darling, CramĂ©r Von Mises and Watson test, for the IG case are discussed. It has to be remarked that this chapter ignores graphical techniques useful for distribution fitting, as \(Q-Q\) plots.

The seventh chapter is the last chapter of the first part. It is the most extensive, and is connected with compound distributions, which may arise, as a result of mixing the IG distribution with some other broadly used distributions such as the Poisson, the normal and the exponential distributions. The resultant compound distributions are very useful whenever one needs to fit a distribution to long-tailed data. In addition, there is a description of the implementation of the mixture of Poisson and IG distributions as an error distribution of a linear regression model, which is appropriate for modeling overdispersed count data.

The second part includes a plethora of applications in several disciplines as diverse as life testing, regression analysis, physiology, entomology, control charts, demography.

The first part of the book, which as previously mentioned, covers theoretical issues, contains a large number of theorems whose proofs require of the reader a strong background on distributions, and statistical theory. Otherwise, its reading becomes hard, in as much as it does not comprise an introductory chapter devoted to the essential theory. On the other hand, the comprehension of the second part of the book is relatively easier. In addition, the second part can be beneficial to many scientists who would wish to utilize the IG distribution in their field of research.

Finally, it has to be noted that the book contains a vast bibliography related to the IG distribution, and can thus serve as a valuable tool for the researcher.

Reviewer: E.Xekalaki (Athens)

### MSC:

62E10 | Characterization and structure theory of statistical distributions |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62F10 | Point estimation |

62G10 | Nonparametric hypothesis testing |

62F03 | Parametric hypothesis testing |