In this paper, we discuss the shape of the hazard function of Birnbaum-Saunders distribution. Specifically, we establish that the hazard function of Birnbaum-Saunders distribution is an upside down function for all values of the shape parameter. In reliability and survival analysis, as it is often of interest to determine the point at which the hazard function reaches its maximum, we propose different estimators of that point and evaluate their performance using Monte Carlo simulations. Next, we analyze a data set and illustrate all the inferential methods developed here and finally make some concluding remarks.

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In several studies in Survival Analysis, the cause of failure/ death of items or individuals may be attributable to more than one cause. In this chapter, we consider the competing risks model when the data is progressively Type-II censored. We provide different techniques for the analysis of the model under the assumption of independent causes of failure and exponential lifetimes. The maximum likelihood estimators of the different parameters and the UMVUE's are obtained. In addition, the exact distributions of the different estimators are derived. We also derive the UMP and UMPU test for the equality of the failure rates of the competing risks.We consider the Bayesian estimation using the Inverse Gamma distribution as a prior. To assess the performance of all these estimators, confidence intervals are developed using the exact, asymptotic, and bootstrap distributions. In the Bayesian context, we develop credible intervals for the parameters. The different methods are compared through a simulation study, and the analysis of a real dataset. Finally, we also provide some insight into inference under the Weibull model and dependent causes of failure.