Cramér-Rao-Leibniz Lower Bound — A new estimation bound for finite support measurement noise

In this paper we introduce a new bound on an estimator´s error, derived from the classical Cramér-Rao Lower Bound (CRLB), for cases where the support of the likelihood function (LF) exhibits parameter-dependence. Parameter-dependent support of the LF arises here when an unknown parameter is observed in the presence of additive measurement noise and the measurement noise pdf has a finite support. This new modified CRLB - designated as the Cramér-Rao-Leibniz Lower Bound (CRLLB), since it relies on Leibniz integral rule - is presented and its use illustrated. The CRLLB is shown to provide, for example, a valid bound for the problem of uniform measurement noise for which the CRLB was shown in the literature as not valid. Furthermore, it is demonstrated that, in light of the CRLLB, the ML estimator in the uniform measurement noise case is statistically efficient, i.e., the estimator´s variance is equal to the CRLLB.