A “Reciprocity” Property of the Unbiased Cramér–Rao Bound for Vector Parameter Estimation

In this letter, we consider Cramér-Rao bounds (CRBs) on the variance of unbiased vector parameter estimators. It is well known that the CRB for the estimation of a parameter α, assuming that a second parameter β is unknown, is not smaller than the CRB computed assuming β 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 α when β is unknown equals the performance loss in estimating β with α unknown. Though this property is of mainly theoretical interest, some examples are given which show that it can be useful also in practical applications.