Estimation after Parameter Selection: Performance Analysis and Estimation Methods

In many practical parameter estimation problems, prescreening and parameter selection are performed prior to estimation. In this paper, we consider the problem of estimating a preselected unknown deterministic parameter chosen from a parameter set based on observations according to a predetermined selection rule, Ψ. The data-based parameter selection process may impact the subsequent estimation by introducing a selection bias and creating coupling between decoupled parameters. This paper introduces a post-selection mean squared error (PSMSE) criterion as a performance measure. A corresponding Cramér-Rao-type bound on the PSMSE of any Ψ-unbiased estimator is derived, where the Ψ-unbiasedness is in the Lehmann-unbiasedness sense. The post-selection maximum-likelihood (PSML) estimator is presented. It is proved that if there exists an Ψ-unbiased estimator that achieves the Ψ-Cramér-Rao bound (CRB), i.e., an Ψ-efficient estimator, then it is produced by the PSML estimator. In addition, iterative methods are developed for the practical implementation of the PSML estimator. Finally, the proposed Ψ-CRB and PSML estimator are examined for exponential populations, a linear Gaussian model applicable in clinical research, and spectrum sensing in cognitive radio communication.