Minimax estimation of large precision matrices with bandable Cholesky factor
Cholesky因子可带状化的大型精度矩阵的极小极大估计AI 解读 ↓
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Cholesky因子可带状化的大型精度矩阵的极小极大估计
- 研究问题
- 当精度矩阵的Cholesky因子具有可带状化结构时，两类参数空间在算子范数和Frobenius范数下的最优估计速率有何差异？
- 方法与思路
- 作者构造极小极大下界，分别采用局部裁剪与回归阈值估计，并考察Lepski自适应及非正态变换模型扩展。
- 材料陈述的贡献
- 摘要称两类参数空间在算子范数下存在根本速率差异，在Frobenius范数下则享有相同最优速率，并提出相应估计程序。
- 阅读关注点
- 应核对两类参数空间、变量排序与谱条件的定义，以及不同范数下的上下界、自适应要求和扩展模型条件。
- 阅读边界
- 仅据OpenAlex索引摘要，稿件版本未注明；具体速率和参数限制未给出，不能把纵向数据应用举例视为一般面板模型结论。
原文摘要
The last decade has witnessed significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the variables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this paper we focus on two commonly used types of parameter spaces and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found. Two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed. For the operator norm our optimal procedure is based on a novel local cropping estimator, targeting on all principle submatrices of the precision matrix, while for the Frobenius norm our optimal procedure relies on a delicate regression-based thresholding rule. Lepski’s method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model. Numerical studies are carried out to confirm our theoretical findings.