This paper addresses the fundamental task of estimating covariance matrix functions for high-dimensional functional data/functional time series. We consider two functional factor structures encompassing either functional factors with scalar loadings or scalar factors with functional loadings, and postulate functional sparsity on the covariance of idiosyncratic errors after taking out the common unobserved factors. To facilitate estimation, we rely on the spiked matrix model and its functional generalization, and derive some novel asymptotic identifiability results, based on which we develop DIGIT and FPOET estimators under two functional factor models, respectively. Both estimators involve performing associated eigenanalysis to estimate the covariance of common components, followed by adaptive functional thresholding applied to the residual covariance. We also develop functional information criteria for the purpose of model selection. The convergence rates of estimated factors, loadings, and conditional sparse covariance matrix functions under various functional matrix norms, are respectively established for DIGIT and FPOET estimators. Numerical studies including extensive simulations and two real data applications on mortality rates and functional portfolio allocation are conducted to examine the finite-sample performance of the proposed methodology.

翻译：本文研究了高维函数数据/函数时间序列中协方差矩阵函数估计的基本问题。我们考虑两种函数因子结构，分别包含函数因子与标量载荷或标量因子与函数载荷，并假设在剔除共同不可观测因子后，异质性误差的协方差具有函数稀疏性。为便于估计，我们借助尖峰矩阵模型及其函数推广，推导了新的渐近可识别性结果，并基于此分别在两种函数因子模型下提出了DIGIT和FPOET估计量。这两种估计量均通过相关特征分析来估计共同成分的协方差，随后对残差协方差进行自适应函数阈值处理。我们还开发了函数信息准则以用于模型选择。对于DIGIT和FPOET估计量，分别建立了在多种函数矩阵范数下估计因子、载荷及条件稀疏协方差矩阵函数的收敛速率。通过包含广泛模拟和两个实际数据分析（死亡率数据与函数投资组合分配）的数值研究，验证了所提方法在有限样本下的表现。