Partial Bayes Estimation of Two Parameter Gamma Distribution Under Non-Informative Prior

In Bayesian analysis, empirical and hierarchical methods are two main approaches for the estimation of the parameter(s) involved in the prior distribution of one parameter. But in the multi-parameter model, e.g., Gamma(α, p), where both the parameters are unknown, idea of the ‘Partial Bayes (PB) Estimation’ is introduced. When we do no have proper belief regarding the joint parameters of the distribution of the variable and when we are estimating one parameter in presence of others, such method may be used. Partial Bayes estimation of the scale parameter p is done by putting the estimate of the another parameter α obtained by some other classical method in case of two parameter Gamma distribution. Using non-informative prior and computing the risk, it is found that the Partial Bayes estimator has less risk than the Bayes estimator. For this, simulation studies for some choices of shape parameter values have been done. In case of the shape parameter, posterior mean and posterior variance are evaluated through simulations to obtain the risk values for estimator of α with known scale parameter. Finally after fifitting this distribution, two real datasets are illustrated to see the performance of the Partial Bayes estimator.