Multivariate Birnbaum-Saunders Distributions: Modelling and Applications

Abstract

Abstract:Since its origins and numerous applications in material science, the Birnbaum-Saunders family of distributions has now found widespread uses in some areas of the applied sciences such as agriculture, environment and medicine, as well as in quality control, among others. It is able to model varied data behaviour and hence provides a flexible alternative to the most usual distributions. The family includes Birnbaum-Saunders and log-Birnbaum-Saunders distributions in univariate and multivariate versions. There are now well-developed methods for estimation and diagnostics that allow in-depth analyses. This paper gives a detailed review of existing methods and of relevant literature, introducing properties and theoretical results in a systematic way. To emphasise the range of suitable applications, full analyses are included of examples based on regression and diagnostics in material science, spatial data modelling in agricultural engineering and control charts for environmental monitoring. However, potential future uses in new areas such as business, economics, finance and insurance are also discussed. This work is presented to provide a full tool-kit of novel statistical models and methods to encourage other researchers to implement them in these new areas. It is expected that the methods will have the same positive impact in the new areas as they have had elsewhere.

Data-Influence Analytics in Predictive Models Applied to Asthma Disease

“…Estimation methods for the regression parameters upon a high censoring may be studied by a mixture structure [53][54][55]. (iii) An extension of the present study to the multivariate case is also of practical relevance [52,56,57].…”

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

“…For the multivariate and matrix‐variate cases, there are open problems, including those on generalized and related distributions (eg, BS‐ t and BS‐elliptical), and estimation methods in both classical and Bayesian approaches, which can be built upon the works of, for example, Cancho et al, Caro‐Lopera et al, Paula et al, Kundu et al, and Aykroyd et al…”

“…An additional note would be on the potential applications of the BS regression, BS‐ACD, and related models. Issues on heteroskedasticity, nonlinearity, time series characteristics, prior information and restrictions, and/or the Bayesian approach involved in model formulation, estimation, inference, and diagnostics are paid extensive attention in, for example, econometrics, finance, and insurance in terms of both research and practice (see, eg, the works of Ahmed et al, Paula et al, Liu et al, Santos‐Neto et al, and Aykroyd et al). It would be useful to obtain further developed methodologies and applications of the BS‐based and related models addressing these mentioned issues.…”

Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications” and a novel multivariate data analytics for an economics example in the textile industry

“…The recent article by Aykroyd et al also includes a review of multivariate Birnbaum‐Saunders distributions and applications. Work on multivariate Birnbaum‐Saunders models and their applications was presented by Marchant et al, and applications of multivariate Birnbaum‐Saunders distributions in spatial modeling appeared in the works of Garcia‐Papani et al In addition, multivariate cumulative damage models and their relationship with time of occurrence in the context of multicomponent systems were recently derived by Fierro et al…”