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**The inverse Gaussian distribution: theory, methodology, and applications.**
*(English)*
Zbl 0701.62009

Statistics: Textbooks and Monographs, 95. New York etc.: Marcel Dekker, Inc. viii, 213 p. (1989).

The Inverse Gaussian Distribution (IGD) and its possible usefulness in statistics came to the attention of statisticians about 15 years ago. The purpose of this book is to summarize as completely as they are presently known the properties of this distribution and its uses in theoretical and applied statistics.

The book is divided into 11 chapters with chapter 1 as an introduction. Chapter 2 gives the properties of the distribution. As this distribution has its origin in the Brownian motion as a first time distribution, a Wiener process with both positive and negative drift is considered in Chapter 3. Wald’s distribution for the sample size in a sequential probability ratio test and the inversion law are also discussed. Chapter 4 gives certain useful transformations and characterizations. Chapter 5 considers maximum likelihood estimators and their distributions, and Chapter 6 gives significance tests with numerical examples.

Chapter 7 considers for certain parameterizations the natural conjugate prior and some diffuse priors under the heading ‘Bayesian Inference’. In Chapter 8, certain regression models for IGD are studied and the possibility of skewed data analysis in a manner similar to that permissible under the usual theory of linear models is explored.

Chapter 9 discusses the estimation of reliability functions and certain tolerance and prediction limits are obtained. In Chapter 10, the authors review practical applications of the IGD - some of which are data oriented. The first example is from cardiology, the second from hydrology, and so on. When considering human behaviour, the use of IGD is justified by goodness-of-fit considerations. The last Chapter attempts to formulate bivariate and multivariate IGD and an inverse Gaussian process.

The book is completed with references, equations, tables and figures and will be useful to statisticians, mathematicians, engineers and others.

The book is divided into 11 chapters with chapter 1 as an introduction. Chapter 2 gives the properties of the distribution. As this distribution has its origin in the Brownian motion as a first time distribution, a Wiener process with both positive and negative drift is considered in Chapter 3. Wald’s distribution for the sample size in a sequential probability ratio test and the inversion law are also discussed. Chapter 4 gives certain useful transformations and characterizations. Chapter 5 considers maximum likelihood estimators and their distributions, and Chapter 6 gives significance tests with numerical examples.

Chapter 7 considers for certain parameterizations the natural conjugate prior and some diffuse priors under the heading ‘Bayesian Inference’. In Chapter 8, certain regression models for IGD are studied and the possibility of skewed data analysis in a manner similar to that permissible under the usual theory of linear models is explored.

Chapter 9 discusses the estimation of reliability functions and certain tolerance and prediction limits are obtained. In Chapter 10, the authors review practical applications of the IGD - some of which are data oriented. The first example is from cardiology, the second from hydrology, and so on. When considering human behaviour, the use of IGD is justified by goodness-of-fit considerations. The last Chapter attempts to formulate bivariate and multivariate IGD and an inverse Gaussian process.

The book is completed with references, equations, tables and figures and will be useful to statisticians, mathematicians, engineers and others.

Reviewer: V.P.Gupta

### MSC:

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

62E10 | Characterization and structure theory of statistical distributions |

62E15 | Exact distribution theory in statistics |

62F15 | Bayesian inference |

62N05 | Reliability and life testing |