Cyclic Misspecified Cramer-Rao Bound for Periodic Parameter Estimation

In many practical parameter estimation problems, the observation model is periodic with respect to the unknown parameters. In these cases, the appropriate estimation criterion is periodic in the parameter space, and cyclic performance bounds should be used. However, existing cyclic performance bounds do not account for the common scenario of model misspecification. The misspecified Cramér-Rao bound (MCRB) provides a lower bound on the mean-squared-error (MSE) for estimation problems under model misspecification. However, the MCRB does not provide a valid bound for periodic problems. In this paper, we close this gap by developing the cyclic MCRB, which is a lower bound on the mean cyclic error (MCE) of any Lehmann unbiased estimator under model misspecification in periodic estimation problems. Thus, it can be seen as a generalization of the cyclic CRB for cases where the assumed model (observations distribution) may be different from the true one. The proposed cyclic MCRB and the performance of the misspecified maximum likelihood (MML) estimator are compared in terms of MCE in direction-of-arrival (DOA) estimation under the misspecified assumption of white additive noise, where the true covariance is colored.