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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.