The Multidimensional Cramér–Rao–Leibniz Lower Bound for Likelihood Functions With Parameter-Dependent Support

One regularity condition for the classical Cramér-Rao lower bound (CRLB) of an unbiased estimator to hold-that the support of the likelihood function (LF) should be independent of the parameter to be estimated-has recently been relaxed to the case of parameter-dependent support as long as the LF is continuous at the boundary of its support. For the case where the LF is not continuous on the boundary of its support, a new modified CRLB-designated the Cramér-Rao-Leibniz lower bound (CRLLB) as it relies on the Leibniz integral rule-has also been presented for the scalar parameter case. The present work derives the multidimensional CRLLB for the case of LF with parameter-dependent support by applying the general Leibniz integral rule to complete the framework of the CRLLB.