Barankin-Type Lower Bound on Multiple Change-Point Estimation

Abstract

We compute lower bounds on the mean-square error of multiple change-point estimation. In this context, the parameters are discrete and the Cramér-Rao bound is not applicable. Consequently, we focus on computing the Barankin bound (BB), the greatest lower bound on the covariance of any unbiased estimator, which is still valid for discrete parameters. In particular, we compute the multi-parameter version of the Hammersley-Chapman-Robbins, which is a Barankin-type lower bound. We first give the structure of the so-called Barankin information matrix (BIM) and derive a simplified form of the BB. We show that the particular case of two change points is fundamental to finding the inverse of this matrix. Several closed-form expressions of the elements of BIM are given for changes in the parameters of Gaussian and Poisson distributions. The computation of the BB requires finding the supremum of a finite set of positive definite matrices with respect to the Loewner partial ordering. Though, each matrix in this set of candidates is a lower bound on the covariance matrix of the estimator, the existence of a unique supremum w.r.t. to this set, i.e., the tightest bound, might not be guaranteed. To overcome this problem, we compute a suitable minimal-upper bound to this set given by the matrix associated with the Lowner-John Ellipsoid of the set of hyper-ellipsoids associated to the set of candidate lower-bound matrices. Finally, we present some numerical examples to compare the proposed approximated BB with the performance achieved by the maximum likelihood estimator.

Performance bounds under misspecification model for MIMO radar application

“…However, the misspecified Barankin bound will still be an interesting tool to analyse asymptotic performance of the mismatch MLE, when the regularity conditions of the MCRB will not be satisfied (i.e. [17]). …”

Weiss-Weinstein bound for change-point estimation

“…, y N ] ∈ Ω ⊂ R M ×N , which can be obtained, for example, from a multiple sensor system. In the context of a single change-point estimation, these observations are modeled as follows [11], [12]:…”

“…To the best of our knowledge, this has been done first in [11] where the Chapman-Robbins bound has been studied. Then, this result has been extended in [12] to the case of multiple change point estimation by using the McAulay-Seidman bound. In both the aforementioned papers, it has been shown that the obtained bounds was quite optimistic with respect to the maximum likelihood estimator empirical MSE.…”

Some results on the Weiss–Weinstein bound for conditional and unconditional signal models in array processing

“…Note that this maximization can be done by using the trace of HG −1 H T or with respect to the Loewner partial ordering [41]. In this paper we will use the trace of HG −1 H T which is enough to obtain tight results.…”

“…In our case of a uniform prior, the results are straightforward and leads to Eqn. (41), (42) and (43).…”