Abstract
The gamma distribution has been extensively used in many areas of applications. In this paper, considering a Bayesian analysis we provide necessary and sufficient conditions to check whether or not improper priors lead to proper posterior distributions. Further, we also discuss sufficient conditions to verify if the obtained posterior moments are finite. An interesting aspect of our findings are that one can check if the posterior is proper or improper and also if its posterior moments are finite by looking directly in the behavior of the proposed improper prior. To illustrate our proposed methodology these results are applied in different objective priors.
Highlights
The Gamma distribution is one of the most well-known distributions used in statistical analysis
We presented a theorem that provides simple conditions under which improper prior yields a proper posterior for the Gamma distribution
An interesting aspect of our findings are that one can check if the posterior is proper or improper and if its posterior moments are finite looking directly at the behavior of the proposed improper prior
Summary
The Gamma distribution is one of the most well-known distributions used in statistical analysis. The joint posterior distribution for α and β, produced by the uniform prior, is π1(α, β|x) βnα Γ(α)n n xiα exp –β i=1 The joint posterior distribution for α and β produced by the Jeffreys rule prior is given by π2(α, β|x) βnα–1 αΓ(α)n n xiα exp –β i=1
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