Log-Weibull extended regression model: Estimation, sensitivity and residual analysis

Abstract

The failure rate function commonly has a bathtub shape in practice. In this paper we discuss a regression model considering new Weibull extended distribution developed by Xie et al. (2002) that can be used to model this type of failure rate function. Assuming censored data, we discuss parameter estimation: maximum likelihood method and a Bayesian approach where Gibbs algorithms along with Metropolis steps are used to obtain the posterior summaries of interest. We derive the appropriate matrices for assessing the local influence on the parameter estimates under different perturbation schemes, and we also present some ways to perform global influence. Also, some discussions on case deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback–Leibler divergence. Besides, for different parameter settings, sample sizes and censoring percentages, are performed various simulations and display and compare the empirical distribution of the Martingale-type residual with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to the martingale-type residual in log-Weibull extended models with censored data. Finally, we analyze a real data set under a log-Weibull extended regression model. We perform diagnostic analysis and model check based on the martingale-type residual to select an appropriate model.