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In many practical parameter estimation problems, such as coefficientestimation of polynomial regression, the true model is unknown and thus, amodel selection step is performed prior to estimation. The data-based modelselection step affects the subsequent estimation. In particular, the oracleCram\'er-Rao bound (CRB), which is based on knowledge of the true model, isinappropriate for post-model-selection performance analysis and system designoutside the asymptotic region. In this paper, we investigatepost-model-selection parameter estimation of a vector with an unknown supportset, where this support set represents the model. We analyze the estimationperformance of coherent estimators that force unselected parameters to zero. Weuse the mean-squared-selected-error (MSSE) criterion and introduce the conceptof selective unbiasedness in the sense of Lehmann unbiasedness. We derive anon-Bayesian Cram\'er-Raotype bound on the MSSE and on the mean-squared-error(MSE) of any coherent estimator with a specific selective-bias function in theLehmann sense. We implement the selective CRB for the special case of sparsevector estimation with an unknown support set. Finally, we demonstrate insimulations that the proposed selective CRB is an informative lower bound onthe performance of the maximum selected likelihood estimator for a generallinear model with the generalized information criterion and for sparse vectorestimation with one step thresholding. It is shown that for these cases theselective CRB outperforms the oracle CRB and Sando-Mitra-Stoica CRB (SMS-CRB)[1].

Cramer-Rao Bound for Estimation After Model Selection and its Application to Sparse Vector Estimation