Case 1: (Generalized Exponential Distribution (GED)) Bayesian case

In our model we utilize certain priors to obtain the Bayesian estimates for the GE distribution in the competing risks setup for complete data. Suppose we have a two component system connected in series. The failure distribution for component 1 and component 2 are given by andrespectively.We assume that the scale parameter () for both the components are same and the shape parameters are  and  for component 1 and component 2 respectively.Let us consider n such systems which are identical to each other.
As usual we assume that are distributed according to GE distribution for i=1,2,…n.

We already know that  and  are the time to failure for the ith system due to component 1 and component 2 respectively and  denote the cause of failure for the i^th system where =1, 2. Then, the time to failure for the ith system will be the minimum of the two,i.e.
The likelihood function for the observed data

*

Since both and are non negative, we assume priors for  to be gamma priors.In many situations, the information about the shape and scale of the sampling distribution is available in an independent manner, and in the similar line we assume that the parameters of are independent a priori.

Hence the joint density function is given by

Therefore the Bayes estimator of any function of , say g() is given by

Since it is not possible to compute the function analytically, we use the Markov Chain Monte Carlo (MCMC) method to approximate this function.