Cramér-Rao Bound for Non-Bayesian Estimation After Detection

We consider a non-Bayesian estimation problem preceded by a detection step. In this framework, the observation model follows one of two hypotheses: an informative hypothesis, where the data contain information about the parameters of interest, and a non-informative hypothesis, where no such information is present. Estimation is only performed if the detection stage declares the informative hypothesis. This setup is common in various signal processing applications, such as communication systems, where decoding occurs only upon message detection, and radar systems, where direction-of-arrival estimation follows target identification. Since the estimation errors are well-defined only when the estimation process occurs, the standard mean-squared error (MSE) criterion is inappropriate for this post-detection estimation problem. To address this, we leverage estimation theory under misspecified models. First, we present the post-selection pseudo-true parameter. Then, we introduce the misspecified post-selection MSE (MPSMSE) criterion, which quantifies the MSE between the estimator and the post-selection pseudo-true parameter conditioning on the event of detecting the informative hypothesis by the considered detector. We developed a new Cramér-Rao-type bound on the MPSMSE that incorporates the knowledge on the detection stage with potential model misspecification. This bound is based on the well-known misspecified Cramér-Rao bound (MCRB), and it provides a robust and accurate benchmark on the MPSMSE for analyzing the performance of practical estimators. Simulations confirm that the proposed bound is a valid lower bound on the MPSMSE of practical estimators and is tighter than existing alternatives.