CRLB for likelihood functions with parameter-dependent support

In this paper we discuss the regularity conditions required for the classical Craḿer-Rao lower bound (CRLB) for real-valued (nonrandom unknown) parameters to hold. It is shown that the commonly assumed requirement that the support of the likelihood function (LF) should be independent of the parameter to be estimated can be replaced by the much weaker requirement that the LF is continuous at the end points of its support. Parameter-dependent support of the LF arises when an unknown parameter is observed in the presence of additive measurement noise and the measurement noise probability density function (pdf) has a finite support. It is also pointed out that the commonly cited requirements of absolute integrability of the derivatives of the LF should be replaced by requirements on the log-LF (LLF). Some practical examples of finite-support measurement noises, which lead to parameter-dependent LF support, are discussed in light of the above. For the case where the LF is not continuous at the end points of its support, a new modified CRLB -designated as the Craḿer-Rao-Leibniz lower bound (CRLLB), since it relies on Leibniz integral rule - is presented and its use illustrated. The CRLLB is shown to provide valid bounds for a number of long-standing problems for which the CRLB was shown in the literature as not valid, in particular, for a uniformly distributed measurement noise.