On Estimation Following Selection with Applications on k-Records and Censored Data

Let X1 and X2 be two independent random variables from gamma populations Pi1,P2 with means alphaθ1 and alphaθ2 respectively, where alpha(> 0) is the common known shape parameter and θ1 and θ2 are scale parameters. Let X(1) ≤ X(2) denote the order statistics ofX1 and X2. Suppose that the population corresponding to the largest X(2) (or the smallest X(1)) observation is selected. The problem ofinterest is to estimate the scale parameters θM (and θJ ) of the selected gamma population under an asymmetric scale invariant loss function.We characterize admissible estimators of θM (or θJ ) within the class of linear estimators of the form cX(2) (or cX(1)). In estimating θM,we derive a minimax estimator and provide sufficient conditions for the inadmissibility of arbitrary invariant estimators of θM. We apply our results to k-Records and censored data. Finally, we extend our results to a subclass of exponential family of distributions.