On the Accuracy and Resolvability of Vector Parameter Estimates

In this paper we address the problem of fundamental limitations on resolution in deterministic parameters estimation. We introduce a definition of resolvability based on probability and incorporating a requirement for accuracy unlike most existing definitions. Indeed in many application the key problem is to obtain distributions of estimates that are not only distinguishable but also accurate and compliant with a required precision. We exemplify the proposed definition with estimators that produce normal estimates, as in the conditional model for which the Gaussianity and efficiency of maximum likelihood estimators (MLEs) in the asymptotic region of operation (in terms of signal-to-noise ratio and/or in large number of snapshots) is well established, even for a single snapshot. In order to measure the convergence in distribution, we derive a simple test allowing to check whether the conditional MLEs operate in the asymptotic region of operation. Last, we discuss the resolution of two complex exponentials with closely spaced frequencies and compare the results obtained with the ones provided by the various statistical resolution limit released in the open literature.