Non-Bayesian Periodic Cramér-Rao Bound

The Cramér-Rao bound (CRB) is one of the most important tools for performance analysis in parameter estimation. In many practical periodic parameter estimation problems, the appropriate criterion is periodic in the parameter space. However, the CRB does not provide a valid lower bound in such problems. In this paper, the periodic CRB for non-Bayesian periodic parameter estimation is derived. The proposed periodic CRB is a lower bound on the mean-square-periodic-error (MSPE) of any periodic-unbiased estimator, where the periodic-unbiasedness is defined by using Lehmann-unbiasedness. It is shown that if there exists a periodic-unbiased estimator which achieves the bound, then the maximum likelihood produces it. The periodic CRB and the performance of some periodic-unbiased estimators are compared in terms of MSPE in a linear, Gaussian problem with modulo measurements and for phase estimation problems.