The Cramér-Rao bound for estimation-after-selection

In many practical parameter estimation problems, a model selection is made prior to estimation. In this paper, we consider the problem of estimating an unknown parameter of a selected population, where the population is chosen from a population set by using a predetermined selection rule. Since the selection step may have an important impact on subsequent estimation, ignoring it could lead to biased-estimation and an invalid Cramér-Rao bound (CRB). In this work, the mean-square-selected-error (MSSE) criterion is used as a performance measure. The concept of Ψ-unbiasedness is introduced for a given selection rule, Ψ, by using the Lehmann-unbiasedness definition. We derive a non-Bayesian Cramér-Rao-type bound on the MSSE of any Ψ-unbiased estimator. The proposed Ψ-CRB is a function of the conditional Fisher information and is a valid bound on the MSSE. Finally, we examine the Ψ-CRB for different selection rules for mean estimation in a linear Gaussian model.