On Maximum Likelihood Estimation in the Presence of Vanishing Information Measure

We analyze the parameter estimation mean square error when the Fisher information measure is zero at some points within the parameter space. At these points the Cramer-Rao lower bound diverges and no unbiased estimator can achieve a finite mean square error. Under mild regularity conditions the maximum likelihood estimator is known to be asymptotically unbiased and therefore lower bounded by the Cramer-Rao lower bound. It is therefore of interest to examine the maximum likelihood estimator performance in the presence of vanishing Fisher information measure. We provide new theoretical and practical results. All results are corroborated by simulations