A survey of some recent results on the CRLB for parameter estimation and its extension

This paper surveys some recent results regarding the Cramér-Rao Lower Bound (CRLB) - the requirements for its existence and its extension to the situation where the parameter's likelihood function (LF) support depends on the parameter to be estimated. In the latter case the conventional CRLB does not hold in general. First we revisit the derivation of the CRLB to elucidate the necessary and sufficient conditions needed for the estimation bound. Following this, the situation of parameter-dependent LF support is considered and the Cramér-Rao-Leibniz Lower Bound (CRLLB) is shown to have an additional term that follows from the Leibniz integral rule, hence the name of this recently obtained bound. This result is the key to provide the estimation bound for the case of uniformly distributed measurement noise over a finite interval. It is also shown that in this case the CRLLB is unattainable. Other examples are also discussed, like the truncated Gaussian measurement noise, in which case the CRLLB is shown to be attained. We also present two examples of apparent “super-efficient estimators” where the estimators' variances are smaller than the conventional CRLB and show that these examples have correct CRLLBs, thus setting straight the myth of “super-efficiency”.