A note on unbiased estimation following selection

In this note, we show that the unbiased estimator of the certain parameter of the selected population does not exist. First, we give a new proof of this fact for the selected normal population, a known result in the literature, which brings out some additional features of the problem. Using a different approach, we then extend the result to some other distributions belonging to a one-parametric exponential family. Some applications are discussed. Whenever an unbiased estimator exists, it is shown to be a function of order statistics.