Bayesian inference for the Birnbaum–Saunders nonlinear regression model

Abstract

Bayesian inference, Birnbaum–Saunders distribution, Fatigue life distribution, Lifetime data, MCMC algorithm, Regression models,

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

“…For different inference‐related issues, one may refer to the works of Ahmed et al, Arellano‐Valle et al, Athayde et al, Audrey et al, Azevedo et al, Balakrishnan et al, Balakrishnan and Zhu, Barros et al, Chang and Tang, Cordeiro et al, Cysneiros et al, Desmond and Yang, Farias and Lemonte, Guo et al, Jeng, Lachos et al, Lemonte, Lemonte and Cordeiro, Lemonte et al, Lemonte and Ferrari, Lemonte and Patriota, Li et al, Li and Xu, Lillo et al, Lu and Chang, Meintanis, Moala et al, Niu et al, Padgett and Tomlinson, Pérez and Correa, Qu and Xie, Riquelme et al, Sánchez et al, Santana et al, Santos‐Neto et al, Saulo et al, Sha and Ng, Teimouri et al, Tsionas, Upadhyay and Mukherjee, Vanegas and Paula, Vanegas et al, Vilca et al, Wang, Wang and Fei, Wang et al, Xiao et al, Xie and Wei, Xu and Tang, Xu et al, Zhu and Balakrishnan, and the references cited therein.…”

“…where g(u), with u > 0, is a real-valued function and corresponds to the kernel of the PDF of X, and c is the normalizing constant, such that f X (x) is a valid PDF. If the random variable X has the PDF in (80), it will be denoted by EC(µ, σ 2 , g). Here, µ and σ are the location and scale parameters, respectively, and g is the corresponding kernel.…”

“…where c is the normalizing constant as mentioned in (80). For different special cases, c can be explicitly obtained and these can be found in Díaz-García and Leiva-Sánchez [69].…”

Improved likelihood-based inference in Birnbaum–Saunders nonlinear regression models

“…In particular, Lemonte and Cordeiro [7] presented a simple matrix formula for the skewness of the distribution of the maximum likelihood estimators (MLEs) of the parameters in this class of models, Lemonte and Patriota [8] developed diagnostic analysis to detect influential observations, and Vanegas et al [9] proposed a general definition of residuals and investigated their statistical properties analytically and empirically. A Bayesian analysis under a normal-gamma prior for the class of BSNLMs was developed by Farias and Lemonte [10] .…”

An extension of Birnbaum-Saunders distributions based on scale mixtures of skew-normal distributions with applications to regression models

“…These data were considered by Farias and Lemonte (2011) in a nonlinear regression model and Santana et al (2011) in a BS linear regression model by using the skew-normal distribution. These data correspond to 46 metal pieces that were tested until failure.…”