Notes on the tightness of the hybrid Cramér-Rao lower bound

In this paper, we study the properties of the hybrid Cramér-Rao bound (HCRB). We first address the problem of estimating unknown deterministic parameters in the presence of nuisance random parameters. We specify a necessary and sufficient condition under which the HCRB of the nonrandom parameters is equal to the Cramér-Rao bound (CRB). In this case, the HCRB is asymptotically tight [in high signal-to-noise ratio (SNR) or in large sample scenarios], and, therefore, useful. This condition can be evaluated even when the CRB cannot be evaluated analytically. If this condition is not satisfied, we show that the HCRB on the nonrandom parameters is always looser than the CRB. We then address the problem in which the random parameters are not nuisance. In this case, both random and nonrandom parameters need to be estimated. We provide a necessary and sufficient condition for the HCRB to be tight. Furthermore, we show that if the HCRB is tight, it is obtained by the maximum likelihood /maximum a posteriori probability (ML/MAP) estimator, which is shown to be an unbiased estimator which estimates both random and nonrandom parameters simultaneously optimally (in the minimum mean-square-error sense).