Estimation and Detection Under Misspecification and Complex Elliptically Symmetric Distributions

Parameter estimationParameter estimation and detectionDetection theory play critical roles in the development of signal and array processing algorithms, especially in applications such as radar, sonar, and wireless communications. Theoretical bounds in detectionDetection and estimation are tied to the true underlying data probability distribution. In practice, however, the assumed data model is often imperfect, i.e., it is commonly misspecified at some level. Firstly, we develop the Cramér–Rao bound (CRB)Cramer-Rao bound (CRB) under model misspecification, including a multivariate generalization of Blyth’s theoremBlyth’s theorem that guarantees an inequality of the Cramér–Rao typeInequality of the Cramer-Rao type (ICRT). Secondly, we explore the classic problem of adaptive radar detectionRadar detection under a class of complex multivariate elliptically symmetric (CMES) distributionsComplex multivariate elliptically symmetric (CMES) distribution known as the compound Gaussian. The generalized likelihood ratio testGeneralized likelihood ratio test (GLRT) (GLRT) is derived along with its finite sample receiver operating characteristicsReceiver operating characteristic (ROC) (ROC). Lastly, the asymptotic (large sample) distribution of the GLRTGeneralized likelihood ratio test (GLRT) in general is explored under model misspecification.