Estimation of the mean of the selected gamma population

Let л₁ and л₂ denote two independent gamma populations G(α₁, p) and G(α₂, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Y_i denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Y_i is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c₁x₍₁₎ +c₁x₍₂₎ and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)^-1x₍₁₎ is minimax and dominates the UMVUE. Also UMVUE is not minimax.