Bayesian cramér-rao type bound for risk-unbiased estimation with deterministic nuisance parameters

In this paper, we derive a Bayesian Cramér-Rao type bound in the presence of unknown nuisance deterministic parameters. The most popular bound for parameter estimation problems which involves both deterministic and random parameters is the hybrid Cramér-Rao bound (HCRB). This bound is very useful especially, when one is interested in both the deterministic and random parameters and in the coupling between their estimation errors. The HCRB imposes locally unbiasedness for the deterministic parameters. However, in many signal processing applications, the unknown deterministic parameters are treated as nuisance, and it is unnecessary to impose unbiasedness on these parameters. In this work, we establish a new Cramér-Rao type bound on the mean square error (MSE) of Bayesian estimators with no unbiasedness condition on the nuisance parameters. Alternatively, we impose unbiasedness in the Lehmann sense for a risk that measures the distance between the estimator and the minimum MSE estimator which assumes perfect knowledge of the nuisance parameters. The proposed bound is compared to the HCRB and MSE of Bayesian estimators with maximum likelihood estimates for the nuisance parameters. Simulations show that the proposed bound provides tighter lower bound for these estimators.