On the asymptotic form of the posterior probability density function for distributions with polynomial failure rates

In this paper, we derive an asymptotic form of the posterior distribution density for parameters of distributions with polynomial failure rates, using a Bayesian approach with a weakly informative prior. The n-dimensional credible region for the distribution parameters is estimated in general form based on the obtained posterior distribution. The order of the polynomial hazard function is determined using two complementary principles: the optimal order minimizes the size of the credible region, while the most probable order is inferred from the structure of the posterior distribution. Parameter estimates are obtained in two ways: as statistical means via the partition function, and as most probable values. The system lifetime is evaluated for both the expected and most probable parameter values. The proposed models are validated for goodness of fit using several examples with data sets from the literature.