Performance Bounds for Parameter Estimation under Misspecified Models: Fundamental Findings and Applications

The objective of this article is to provide an accessible and, the same time, comprehensive treatment of the fundamental concepts about CRBs and efficient estimators in the presence of model misspecification. Every SP practitioner is aware of the fact that, in almost all practical applications, a certain amount of mismatch between the true and the assumed statistical data models is inevitable. Despite its ubiquity, the assessment of performance bounds under model misspecification appears to have received limited attention from the SP community, while it has been deeply investigated by the statistical community. The first aim of this tutorial is to propose to wide SP audience a comprehensive review of the main contributions to the mismatched estimation theory, both for the deterministic and Bayesian frameworks, with a particular focus on the derivation of CRB under model mismatching. Specifically, we have described how the classical tools of the estimation theory can be generalized to address a mismatched scenario. First, the MCRB has been introduced and the behavior of the MML estimator investigated. Second, results related to the deterministic estimation framework have been extended to the Bayesian one. The existence and the asymptotic properties of a MB estimator have been discussed. Moreover, some general ideas about the possibility to derive MBCRBs have been provided. In the last part of the article, we showed how to apply the theoretical findings to two well-known relevant problems: the DOA estimation in array processing and the estimation of the disturbance covariance matrix for adaptive radar detection.