CRAMR-RAO BOUND FOR ESTIMATION AFTER MODEL SELECTION AND ITS APPLICATION TO SPARSE VECTOR ESTIMATION

In many practical parameter estimation problems, such as coefficient estimation of polynomial regression, the true model is unknown and thus, a model selection step is performed prior to estimation. The data-based model selection step affects the subsequent estimation. In particular, the oracle Cram�r-Rao bound (CRB), which is based on knowledge of the true model, is inappropriate for post-model-selection estimation. In this paper, we investigate post-model-selection parameter estimation of a vector with an unknown support set, where this support set represents the model. We analyze the estimation performance of coherent estimators that force unselected parameters to zero. We use the mean-squared-selected-error (MSSE) criterion and introduce the selective unbiasedness in the sense of Lehmann unbiasedness. We derive a non-Bayesian Cram�r-Rao-type bound on the MSSE and on the mean-squared-error (MSE) of any coherent estimator with a specific selective-bias function. Finally, we demonstrate in simulations that the proposed selective CRB is an informative lower bound on the performance of the maximum selected likelihood estimator for a general linear model with the generalized information criterion and for sparse vector estimation with one-step thresholding. It is shown that for these cases the selective CRB outperforms the oracle CRB and Sando-Mitra-Stoica CRB.