The Misspecified Cramer-Rao Bound and Its Application to Scatter Matrix Estimation in Complex Elliptically Symmetric Distributions

Abstract

This paper focuses on the application of recent results on lower bounds under misspecified models to the estimation of the scatter matrix of complex elliptically symmetric (CES) distributed random vectors. Buildings upon the original works of Q. H. Vuong [Cramér-Rao Bounds for Misspecified Models, Div. of the Humanities and Social Sci., California Inst. of Technol., Pasadena, CA, USA, Working Paper 652, Oct. 1986] and Richmond-Horowitz [“Parameter Bounds on Estimation Accuracy Under Model Misspecification,” IEEE Trans. Signal Process., vol. 63, no. 9, pp. 2263-2278, May 2015], a lower bound, named misspecified Cramér-Rao bound (MCRB), for the error covariance matrix of any unbiased (in a proper sense) estimator of a deterministic parameter vector under misspecified models, is introduced. Then, we show how to apply these results to the problem of estimating the scatter matrix of CES distributed data under data mismodeling. In particular, the performance of the maximum likelihood (ML) estimator are compared, under mismatched conditions, with the MCRB and with the classical CRB in some relevant study cases.