Revisit of relationships and models for the Birnbaum-Saunders and inverse-Gaussian distributions

Abstract

The Birnbaum-Saunders distribution was derived in 1969 as a lifetime model for a specimen subjected to cyclic patterns of stresses and strains, and the ultimate failure of the specimen is assumed to be due to the growth of a dominant crack in the material. The inverse Gaussian distribution is used to describe the first passage time for a particle (moving with constant velocity) that is subject to linear Brownian motion. These two models have a rich history, and they have been shown to be much related. In this article, these two models will be reviewed and comparisons will be made. Specifically, two moment-ratio diagrams will be presented that gives insight to the reason of both distributions often achieve similar fits to experimental data. Next, a generalized Birnbaum-Saunders distribution, will be presented and several properties will be derived. In particular, it will be shown that this generalized model can be expressed as a mixture of inverse Gaussian-type random variables (similar to the two-parameter Birnbaum-Saunders model). Estimation of the parameters in the generalized Birnbaum-Saunders distribution will be discussed. Lastly, some conclusions from this investigation are presented.

A new log-linear bimodal Birnbaum–Saunders regression model with application to survival data

“…The chief goal of our paper is to propose a log-linear regression model based on a bimodal version of the Birnbaum-Saunders distribution that has been recently introduced by Owen and Ng (2015). Such a distribution is more flexible than the original BS law and we use it as the basis for developing a regression model that is more general than the BSRM introduced by Rieck and Nedelman (1991).…”

“…They obtained it by adding a second shape parameter to the BS distribution function. More recently, Owen and Ng (2015) analyzed the distribution, which the authors denoted by GBS 2 . In particular, they investigated the relationships among the inverse Gaussian, BS and GBS 2 distributions.…”

“…Several useful properties of the GBS 2 distribution were obtained by Owen and Ng (2015), some of them also holding for the BS distribution. For instance, it was shown that η is the distribution median, which can be easily verified from Equation (1).…”

Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications”

“…Desmond observed that the hazard functions of the BS and IG distributions are very similar and came up with a similar conclusion as Bhattacharyya and Fries that the IG distribution is preferred for statistical analysis. For a detailed review of the relationships between the BS and IG distributions, one can refer to the work of Owen and Ng …”

A New Inference Approach for Type-II Generalized Birnbaum-Saunders Distribution

“…Ref. [19] discussed the likelihood-based estimation of parameters and provided interval estimation based on "observed" Fisher's information. In this article, to contribute to the relatively neoteric body of research, we propose a new inference method for the parameter estimation and hypothesis testing for the GBS-II model.…”