Barankin-Type Lower Bound on Multiple Change-Point Estimation
Barankin-Type Lower Bound on Multiple Change-Point Estimation
复制标题
多变点估计的 Barankin 型下界
DOI：
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发表时间：
2010
影响因子：
5.4
通讯作者：
A. Nehorai
中科院分区：
文献类型：
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作者：
P. S. Rosa;A. Renaux;C. Muravchik;A. Nehorai
我们计算了多变点估计均方误差的下界。在这种情况下，参数是离散的，cram<s:1> r- rao界限不适用。因此，我们的重点是计算Barankin界（BB），任何无偏估计量的协方差的最大下界，它仍然对离散参数有效。特别地，我们计算了Hammersley- Chapman-Robbins的多参数版本，它是一个barankin型下界。我们首先给出了所谓的Barankin信息矩阵（BIM）的结构，并推导了BB的简化形式。我们证明了两个变换点的特殊情况是求这个矩阵逆的基础。针对高斯分布和泊松分布参数的变化，给出了BIM构件的几种封闭表达式。BB的计算要求找到一个有限正定矩阵集关于Loewner偏序的极值。虽然这个候选集合中的每个矩阵都是估计量协方差矩阵的下界，但不能保证这个集合存在唯一的上界w.r.t.，即最紧界。为了克服这个问题，我们计算了候选下界矩阵集的超椭球集的Loewner-John椭球集的矩阵给出的这个集合的合适的最小上界。最后，我们给出了一些数值例子来比较所提出的近似BB与最大似然估计的性能。
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. Although 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 Loewner-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.