Discussion of “Birnbaum-Saunders distribution: A review of models, analysis, and applications” and a novel financial extreme value data analytics from natural disasters - Leiva, Víctor (2019)
[Excerpt] The univariate Birnbaum-Saunders (BS) distribution was first postulated to model failure times in material science (see the work of Birnbaum and Saunders1). In this modeling, a cumulative damage exceeds a threshold to produce the failure (see the work of Leiva and Saunders 2). The univariate BS distribution is unimodal, positively skewed (although close to a symmetric distribution in some cases), supported over a positive range of values and possessor of diverse and attractive properties (see the book of Johnson et al 3). The univariate BS distribution has been extensively studied and applied (see the works of Leiva et al 4,5 ). Most of its mathematical and statistical results until 2016 were published in the book of Leiva.6 Multivariate BS distributions were derived as a natural extension to the univariate case, based on mathematical methods, with no fatigue theoretical arguments, different to the univariate BS distribution. Aykroyd et al7 published recently a review on multivariate BS distributions with some applications. In addition, cumulative damage models and their relation to times of occurrence were recently modeled in a multivariate setting for multicomponent systems by Fierro et al.8 Balakrishnan and Kundu9 conducted a complete and interesting review of the BS distribution, which considered phys ical justifications, mathematical and statistical issues, shape analysis and links to other models, as well as formulations and generalizations for the univariate case. Furthermore, extensions to multivariate and matrix-variate versions of the BS distribution were also included. This review provides a full and updated list of references on the topic. However, in the book of Leiva6 and in the review of Balakrishnan and Kundu,9 no attention was paid to an extreme value BS (EVBS) dis tribution, which has several attractive properties and a different conception with respect to its standard version. Indeed, the standard BS distribution cannot be obtained as a particular case of the EVBS distribution, as it happens with other generalizations and extensions of the BS distribution. [...]