Weiss-Weinstein bound on multiple change-points estimation
Résumé
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) which 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 i) Gaussian observations with changes in the mean and/or the variance, and ii) changes in the mean rate of a Poisson distribution. Furthermore, a semi-definite 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 hyper-ellipsoid elements which 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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