Selective Cramér-Rao Bound for Estimation after Model Selection

In many practical parameter estimation problems, such as direction-of-arrival (DOA) estimation, model selection is done prior to estimation. The data-based model selection step affects the subsequent estimation, which may results in a biased estimation and an invalid Cramér-Rao bound (CRB). Additionaly, estimators after model selection are usually assumed to be coherent with the model selection step, such that the deselected parameters are set to zero. In this paper, we show that for coherent estimators an appropriate estimation performance measure is the mean-squared-selected-error (MSSE) criterion. We introduce the concept of selective unbiasedness by using the Lehmann unbiasedness definition. We derive a non-Bayesian Cramér-Rao-type bound on the MSSE of any coherent and selective unbiased estimator. Finally, we demonstrate that the proposed selective CRB (sCRB) is a valid and informative lower bound on the performance of the post-model selection maximum likelihood estimator for linear regression with the Akaikes Information Criterion (AIC) of model selection.