Mixture of the inverse Rayleigh distribution: Properties and estimation in a Bayesian framework

An engineering process is an output from a set of combined processes which may be homogeneous or heterogeneous. To study the lifetime of such processes, we need a model which can accommodate the nature of such processes. Single probability models are not capable of capturing the heterogeneity of nature. However, mixture models of some suitable lifetime distributions, have the potential to highlight such interesting feature. Due to time and cost constraint, in the most lifetime testing experiments, censoring is an unavoidable feature of most lifetime data sets. This article deals with the modeling of the heterogeneity existing in the lifetime processes using the mixture of the inverse Rayleigh distribution, and the spotlight is the Bayesian inference of the mixture model using non-informative (the Jeffreys and the uniform) and informative (gamma) priors. We are considering this particular distribution due to two reasons; the first one is due to its skewed behavior, i.e. in engineering processes, an engineer suspects that high failure rate in the beginning, but after continuous inspection, the failure goes down and the second reason is due to its vast application in many applied fields. A Gibbs sampling algorithm based on adaptive rejection sampling is designed for the posterior computation. A detailed simulation study is carried out to investigate the performance of the estimators based on different prior distributions. The posterior risks are evaluated under the squared error, the weighted, the quadratic, the entropy, the modified squared error and the precautionary loss functions. Posterior risks of the Bayes estimates are compared to explore the effect of prior information and loss functions.