Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications” and a novel financial extreme value data analytics from natural disasters

Abstract

The univariate Birnbaum-Saunders (BS) distribution was first postulated to model failure times in material science (see the work of Birnbaum and Saunders 1 ). 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 al 7 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 Kundu 9 conducted a complete and interesting review of the BS distribution, which considered physical 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 Leiva 6 and in the review of Balakrishnan and Kundu, 9 no attention was paid to an extreme value BS (EVBS) distribution, 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.There are several reasons to justify the use of the BS distribution in the modeling of data with heavy tails. In particular, the EVBS distribution developed by Ferreira et al 10 provides some of these justifications. The EVBS distribution is based on the generalized extreme value (GEV) distribution. As mentioned, the standard BS distribution was derived to solve a problem of material fatigue, but the BS distribution has been also used to solve problems in areas as diverse as environment, finance, and medicine, where heavy-tailed phenomena are often detected. This kind of phenomena is frequently described by extreme value models. Then, the EVBS distribution can be a good alternative to BS distributions because of its good properties to model heavy-tailed phenomena.We discuss some aspects of the BS distribution reviewed by Balakrishnan and Kundu 9 and delve into the EVBS distribution, considering its features, modeling properties, and ne...

The PORTSEA (Portuguese School of Extremes and Applications) and a few personal scientific achievements

“…Then, the BS model can be generalized by switching the distribution of the basis variable using diverse arguments allowing to construct more general classes of models. The development of research work with Victor Leiva, from Pontificia Universidad Catolica de Valparaiso, led us to consider Birnbaum-Saunders' EV laws, studied from different perspectives in the following international articles: [90,194,195,298,299].…”

“…Among many other articles involving this topic, I mention the following ones: [253], with the introduction of non-classical EV models and their application to climatological data; [141], with developments in the estimation of parameters of rare events in environmental time series. And there are some additional international articles on this topic: [311] advanced with an assessment of the extreme limits of nitriding in aluminium extrusion; [213] outlined a refined method for estimating a light right tail, with applications to environmental data; [214] developed different methods for estimating parameters of extreme events based on resampling methods and applied those methods to environmental and financial data; [298] applied Birnbaum-Saunders regression models to environmental data (for a review of this type of models, applied to various data related to natural disasters, see [299]).…”