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 model selection with theoretical guarantees. The convergence rates of involved estimated quantities are respectively established for DIGIT and FPOET estimators. Numerical studies including extensive simulations and a real data application on functional portfolio allocation are conducted to examine the finite-sample performance of the proposed methodology.

翻译：本文针对高维函数数据/函数时间序列的协方差矩阵函数估计这一基本任务展开研究。我们考虑两种函数因子结构，分别包含具有标量载荷的函数因子和具有函数载荷的标量因子，并假设在剔除不可观测的公共因子后，特质误差的协方差具有函数稀疏性。为便于估计，我们基于尖峰矩阵模型及其函数推广，推导出若干新颖的渐近可识别性结果。在此基础上，我们分别在两种函数因子模型下提出了DIGIT和FPOET估计量。两种估计量均涉及通过特征分析估计公共成分的协方差，随后对残差协方差应用自适应函数阈值处理。我们还开发了具有理论保证的模型选择函数信息准则。针对DIGIT和FPOET估计量，我们分别建立了相关估计量的收敛速率。通过包含大量模拟实验和函数投资组合配置实际数据应用的数值研究，检验了所提方法的有限样本性能。