Advances on the Birnbaum-Saunders distribution - Nakamura, Luiz Ricardo (2016)
The Birnbaum-Saunders (BS) distribution is the most popular model used to describe lifetime process under fatigue. Throughout the years, this distribution has received a wide ranging of applications, demanding some more flexible extensions to solve more complex problems. One of the most well-known extensions of the BS distribution is the generalized Birnbaum- Saunders (GBS) family of distributions that includes the Birnbaum-Saunders special-case (BSSC) and the Birnbaum-Saunders generalized t (BSGT) models as special cases. Although the BS-SC distribution was previously developed in the literature, it was never deeply studied and hence, in this thesis, we provide a full Bayesian study and develop a tool to generate random numbers from this distribution. Further, we develop a very flexible regression model, that admits different degrees of skewness and kurtosis, based on the BSGT distribution using the generalized additive models for location, scale and shape (GAMLSS) framework. We also introduce a new extension of the BS distribution called the Birnbaum-Saunders power (BSP) family of distributions, which contains several special or limiting cases already published in the literature, including the GBS family. The main feature of the new family is that it can produce both unimodal and bimodal shapes depending on its parameter values. We also introduce this new family of distributions into the GAMLSS framework, in order to model any or all the parameters of the distribution using parametric linear and/or nonparametric smooth functions of explanatory variables. Throughout this thesis we present five different applications in real data sets in order to illustrate the developed theoretical results.