A "Reciprocity" Property of the Unbiased Cramer-Rao Bound for Vector Parameter Estimation

In this letter, we consider Cramer-Rao bounds (CRBs) on the variance of unbiased vector parameter estimators. It is well known that the CRB for the estimation of a parameter alpha, assuming that a second parameter beta is unknown, is not smaller than the CRB computed assuming beta known. The performance loss can be measured by the ratio between these two bounds. In this work, we derive a "reciprocity" property of the CRB for vector parameter estimation, which indicates that the performance loss in estimating alpha when beta is unknown equals the performance loss in estimating beta with alpha unknown. Though this property is of mainly theoretical interest, some examples are given which show that it can be useful also in practical applications.