Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness
Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness
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DOI：
10.1016/j.jmva.2016.04.001
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发表时间：
2016-07-01
影响因子：
1.6
通讯作者：
Trushin, D.
中科院分区：
文献类型：
--
作者：
Soloveychik, I.;Trushin, D.
我们研究高斯和鲁棒协方差估计，假设真协方差矩阵是两个低维方阵的Kronecker积。在这两种情况下，我们将估计量定义为约束最大似然规划的解。在鲁棒情况下，我们将泰勒估计量定义为球面上某一分布的极大似然估计量。我们给出了估计存在唯一性的严格充分条件，并证明了在均值未知的高斯情形下，p/q + q/p + 2个样本几乎足以保证估计的存在唯一性，其中p和q是Kronecker积因子的维数。在均值已知的鲁棒情况下，相应的足够样本数为max[p/q， q/p] +1。(C) 2016 Elsevier Inc.版权所有。
We study the Gaussian and robust covariance estimation, assuming the true covariance matrix to be a Kronecker product of two lower dimensional square matrices. In both settings we define the estimators as solutions to the constrained maximum likelihood programs. In the robust case, we consider Tyler's estimator defined as the maximum likelihood estimator of a certain distribution on a sphere. We develop tight sufficient conditions for the existence and uniqueness of the estimates and show that in the Gaussian scenario with the unknown mean, p/q + q/p + 2 samples are almost surely enough to guarantee the existence and uniqueness, where p and q are the dimensions of the Kronecker product factors. In the robust case with the known mean, the corresponding sufficient number of samples is max[p/q, q/p] +1. (C) 2016 Elsevier Inc. All rights reserved.