The concept of Statistical Resolution Limit (SRL), which is defined as the minimal separation to resolve two closely spaced signals, is an important tool to quantify performance in parametric estimation problems. This paper generalizes the SRL based on the Cramér-Rao bound to multiple parameters of interest per signal and for multiple signals. We first provide a fresh look at the SRL in the sense of Smith's criterion by using an adequate changes of variable formula. Second, based on the Minkowski distances, we extend this criterion to the important case of multiple parameters of interest per signal and to multiple signals. The results presented herein can be applied to any estimation problem and are not limited to source localization problem.
In the context of multiple change-points estimation, performance analysis of estimators such as the maximum likelihood is often difficult to assess since the regularity assumptions are not met. Focusing on the estimators variance, one can, however, use lower bounds on the mean square error. In this paper, we derive the so-called Weiss-Weinstein bound (WWB) that is known to be an efficient tool in signal processing to obtain a fair overview of the estimation behavior. Contrary to several works about performance analysis in the change-point literature, our study is adapted to multiple changes. First, useful formulas are given for a general estimation problem whatever the considered distribution of the data. Second, closed-form expressions are given in the cases of Gaussian observations with changes in the mean and/or the variance, and changes in the mean rate of a Poisson distribution. Furthermore, a semidefinite programming formulation of the minimization procedure is given in order to compute the tightest WWB. Specifically, it consists of finding the unique minimum volume covering the set constituted by hyperellipsoid elements that are generated using the derived candidate WWB matrices w.r.t. the so-called Loewner partial ordering. Finally, simulation results are provided to show the good behavior of the proposed bound.
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