Optimal rules and robust Bayes estimation of a Gamma scale parameter

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$cX\,{+}\,d$$</EquationSource> </InlineEquation>, when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$X\sim \varGamma (\alpha ,\eta )$$</EquationSource> </InlineEquation>. In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail. Copyright Springer-Verlag 2013