Misspecified Cramer-Rao Bound Research Articles

Misspecified Cramer–Rao Bounds for Blind Channel Estimation Under Channel Order Misspecification

In estimation, the misspecified Cramer–Rao bound (MCRB), which is an extension of the well-known Cramer–Rao bound (CRB) when the underlying system model is misspecified, has recently attracted much attention. In this paper, we introduce a new interpretation of the MCRB, called the generalized MCRB (GMCRB), via the Moore–Penrose inverse operator. This bound is useful for singular problems and particularly blind channel estimation problems in which the Hessian matrix is noninvertible. Two closed-form expressions of the GMCRB are derived for unbiased blind estimators when the channel order is misspecified. The first bound deals with deterministic models where both the channel and unknown symbols are deterministic. The second one is devoted to stochastic models where we assume that transmitted symbols are unknown random variables i.i.d. drawn from a Gaussian distribution. Two case studies of channel order misspecification are investigated to demonstrate the effectiveness of the proposed GMCRBs over the classical CRBs. When the channel order is known or accurately estimated, both generalized bounds reduce to the classical bounds. Besides, the stochastic GMCRB is lower than the deterministic one, especially at high SNR.

The Constrained Misspecified Cramér–Rao Bound

The aim of this letter is to provide a constrained version of the misspecified Cramer–Rao bound (MCRB). Specifically, the MCRB is a lower bound on the error covariance matrix of any unbiased (in a proper sense) estimator of a deterministic parameter vector under misspecified models, i.e., when the true and the assumed data distributions are different. Here, we aim at finding an expression of the MCRB for estimation problems involving continuously differentiable equality constraints. Our proof generalizes the derivation of the classical constrained CRB (CCRB) by showing that the constrained MCRB (CMCRB) can be obtained by exploiting the building blocks of its unconstrained counterpart and a basis of the null space of the constraint’s Jacobian matrix. The conditions for the existence of the CMCRB are also discussed.

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

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 [Cramer-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 Cramer–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.