Precision Matrix Estimation With ROPE

It is known that the accuracy of the maximum likelihood-based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article, we propose a ridge-type operator for the precision matrix estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context. The performance of the proposed method is illustrated by a simulation study and real data applications. Computer code used in the example analyses as well as other supplementary materials for this article are available online.

众所周知，通过惩罚对数似然估计，可以提升基于极大似然的协方差与精度矩阵估计的精度。本文提出一种用于精度矩阵估计的岭型算子（简记为ROPE），以弗罗贝尼乌斯范数（Frobenius norm）作为惩罚函数来最大化惩罚似然函数。研究表明，当使用惩罚对数似然时，精度矩阵的收缩估计量存在显式闭形式表达式，该形式与回归场景下的岭回归具有相似性。本文通过模拟研究与实际数据应用验证了所提方法的性能。本文示例分析所用的计算机代码及其他补充材料均可在线获取。