On the posterior moments for truncation parameter distributions and identihability by posterior mean for exponential distribution with location parameter

The exact analytical expressions of posterior moments for the two general truncation parameter likelihood functions with arbitrary prior distributions are obtained by using the sufficient statistics for these truncation parameter distributions. In particular, the explicit forms for posterior mean and variance are given. Some examples are also discussed to illustrate the obtained results. Next, an explicit expression for mixing distribution in terms of posterior mean for exponential distribution with location parameter is obtained and then the identihability by posterior mean is established. As an example, the gamma mixing distribution family is used to illustrate the results obtained.