Estimating the Parameter of a Selected Population

Abstract

Suppose independent samples from k populations with unknown parameters are taken and one of the populations is selected based on tho data and a prespecified rule. The problem is to estimate the parameter of the selected population. The estimand, G, is a random quantity which depends on both the data and the unknown parameters. While standard estimation methods are inadequate for estimating G, they can be used to estimate the expected value of G. It is shown that the uniformly minimum variance unbiased estimator of E(G) is also the uniformly minimum mean squared error unbiased estimator of G, if the selection rule depends on the data only through a complete sufficient statistic. An approach based on conditional unbiasedness is also discussed.