Cyclic Cramér-Rao-type bounds for periodic parameter estimation

Cyclic lower bounds establish fundamental limits in parametric statistical models with a periodic nature. Non-Bayesian lower bounds on the mean-cyclic-error (MCE) of any cyclic-unbiased estimator, in the Lehmann sense, are useful for performance analysis and system design in periodic estimation. In this paper, we derive two cyclic versions of the Cramér-Rao bound (CRB) based on Mardia's circular information inequality. The proposed cyclic CRBs are lower bounds on the MCE of any cyclic-unbiased estimator. One of these bounds has been recently developed by using the cyclic Hammersley-Chapman-Robbins lower bounds on the MCE. The derivations presented in this paper relates between the cyclic CRB and existing results from directional statistics on manifolds. The properties of the proposed cyclic CRBs are examined. In particular, it is shown that the cyclic CRBs are always lower than the convectional CRB and that the two cyclic CRBs are asymptotically achievable by the performance of the maximum likelihood (ML) estimator. The proposed cyclic CRBs and the performance of the ML estimator are compared in terms of MCE in estimation of the mean of von Mises distributed measurements and for phase estimation with additive white Gaussian noise (AWGN).