Finite mixture of gamma distributions: A conjugate prior

A finite mixture of gamma distributions [Finite mixture of certain distributions. Comm. Statist. Theory Methods 31(12), 2123–2137] is used as a conjugate prior, which gives a nice form of posterior distribution. This class of conjugate priors offers a more flexible class of priors than the class of gamma prior distributions. The usefulness of a mixture gamma-type   prior and the posterior of uncertain parameters λλ for the Poisson distribution are illustrated by using Markov Chain Monte Carlo (MCMC), Gibbs sampling approach, on hierarchical models. Using the generalized hypergeometric function, the method to approximate maximum likelihood estimators for the parameters of Agarwal and Al-Saleh [Generalized gamma type distribution and its hazard rate function. Comm. Statist. Theory Methods 30(2), 309–318] generalized gamma-type distribution is also suggested.