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.

We compute the Weiss-Weinstein bound in the context of change-point estimation in a multivariate time series whatever the considered distribution of the data as well the prior. Closed-form expressions are then given in the case of Gaussian observations with change of mean and variance and in the case of parameter change in a Poisson distribution. The proposed bound is shown to be tighter than the previous bounds which were originally derived in the deterministic context and provides a better approximation of the maximum a posteriori estimator global mean square error.

This paper derives lower bounds for the mean square errors of parameter estimators in the case of Poisson distributed data subjected to multiple abrupt changes. Since both change locations (discrete parameters) and parameters of the Poisson distribution (continuous parameters) are unknown, it is appropriate to consider a mixed Cramér-Rao/Weiss-Weinstein bound for which we derive closed-form expressions and illustrate its tightness by numerical simulations.