A Bayesian lower bound for parameter estimation of Poisson data including multiple changes

Abstract

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.

A Hybrid Lower Bound for Parameter Estimation of Signals With Multiple Change-Points

“…In this sense, the bound derived in this paper is more general -for applications to specific usual distributions, see Section IV-B. Another difference is the estimation framework: while in [19] we used a fully Bayesian point of view, we now consider a hybrid context, in the sense that the parameter vectors η q stacked in η are assumed unknown and deterministic, with true values η (accordingly, the true value of the full parameter vector is denoted by η ), and the parameter vector t is assumed random. Consequently, the estimator (x) is hybrid as well, for example it can be the ML-MAP estimator (Maximum Likelihood-Maximum A Posteriori) [20, p. 12], [21].…”

“…As explained in Section I, we provided in [18] a lower bound on the estimation error for the vector t only (parameter vectors η q were assumed to be known), assuming t is a random vector (Bayesian lower bound). In [19], addressing a specific type of data (namely, a time-series with Poisson entries), the (scalar) parameters η q were assumed unknown and random, i.e., a fully Bayesian point of view for the estimation of t and η was considered. In this paper, we fill the gap between [18] and [19] by generalizing the work presented in [19] to any pre-specified signal distribution f(.…”

“…In [19], addressing a specific type of data (namely, a time-series with Poisson entries), the (scalar) parameters η q were assumed unknown and random, i.e., a fully Bayesian point of view for the estimation of t and η was considered. In this paper, we fill the gap between [18] and [19] by generalizing the work presented in [19] to any pre-specified signal distribution f(. ; η q ) (parameterized by unknown vectors η q ; see Section IV-A).…”

“…To achieve this, we propose the "Hybrid Cramér-Rao-Weiss-Weinstein bound" (HCRWWB, later abbreviated as HB for "hybrid bound", for sake of simplicity), in which the CRB part is associated with the continuous parameters, and the WWB part is associated with the discrete parameters. Note that a Bayesian Cramér-Rao-Weiss-Weinstein bound was proposed in [19] in a fully Bayesian context. In contrast to this bound, whose determination required a numerical integration, we investigate in this paper an HB, which can be determined using closed-form expressions.…”

Performance Bounds for Change Point and Time Delay Estimation

“…The Chapman-Robbins bound has been derived in [14] in the context of one change point and extended to the multiple change point problem in [15]. In the Bayesian context, the Weiss-Weinstein bound has been studied in [16]- [18]. In this paper, we stay in the non-Bayesian context to study the Barankin (or McAulay-Seidman bound) [19], [20] for a change point estimation problem when (contrary to previous works) two sets of non synchronized data are available.…”