Birnbaum–Saunders Nonlinear Regression Model: A Full Bayesian Analysis

The family of distributions proposed by Birnbaum and Saunders (1969, A new family of life distributions, Journal of Applied Probability) can be used to model lifetime data and it is widely applicable to model failure times of fatiguing materials. In this paper, we develop a Bayesian analysis for the Birnbaum–Saunders nonlinear regression model, recently introduced by Lemonte and Cordeiro (2009, Birnbaum–Saunders nonlinear regression models. Computational Statistics and Data Analysis). We have considered a Bayesian analysis based on three different prior specifications. Two different Jeffreys priors and a non-informative prior. Due to the complexity of the model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. We describe tools for model determination, which include the conditional predictive ordinate, the logarithm of the pseudo-marginal likelihood and the pseudo-Bayes factor. Additionally, case deletion influence diagnostics is developed for the joint posterior distribution based on the Kullback–Leibler divergence. The developed procedures are illustrated with an application to a real data set.