Title: Yule-Walker Estimation for Functional Time Series in Hilbert Space Show Chinese title

Title: Hilbert空间中函数时间序列的Yule-Walker估计

Abstract: Recent advances in data collection technologies have led to the widespread availability of functional data observed over time, often exhibiting strong temporal dependence. However, existing methodologies typically assume independence across functions or impose restrictive low-order dependence structures, limiting their ability to capture the full dynamics of functional time series. To address this gap, we investigate higher-order functional autoregressive (FAR) models in Hilbert spaces, focusing on the statistical challenges introduced by infinite dimensionality. A fundamental challenge arises from the ill-posedness of estimating autoregressive operators, which stems from the compactness of the autocovariance operator and the consequent unboundedness of its inverse. We propose a regularized Yule-Walker-type estimation procedure, grounded in Tikhonov regularization, to stabilize the estimation. Specializing to $L^2$ spaces, we derive explicit and computationally feasible estimators that parallel classical finite-dimensional methods. Within a unified theoretical framework, we study the asymptotic properties of the proposed estimators and predictors. Notably, while the regularized predictors attain asymptotic normality, the corresponding estimators of the autoregressive operators fail to converge weakly in distribution under the operator norm topology, due to the compactness of the autocovariance operator. We further analyze the mean squared prediction error (MSPE), decomposing it into components attributable to regularization bias, truncation, and estimation variance. This decomposition reveals the advantages of our approach over traditional linear truncation schemes. Extensive simulations and an application to high-frequency wearable sensor data demonstrate the practical utility and robustness of the proposed methodology in capturing complex temporal structures in functional time series. Show Chinese abstract

Abstract: 数据收集技术的最新进展导致了广泛可用的时间观察到的功能数据，这些数据通常表现出强烈的时间依赖性。 然而，现有的方法通常假设函数之间相互独立，或者施加限制性的低阶依赖结构，限制了它们捕捉功能时间序列完整动态的能力。 为了解决这一差距，我们研究了Hilbert空间中的高阶功能自回归（FAR）模型，重点是无限维数引入的统计挑战。 一个基本的挑战来自于估计自回归算子的不适定问题，这源于自协方差算子的紧致性和其逆的无界性。 我们提出了一种基于Tikhonov正则化的正则化Yule-Walker型估计程序，以稳定估计。 专门针对 $L^2$ 空间，我们推导出明确且计算可行的估计器，这些估计器与经典的有限维方法平行。 在一个统一的理论框架内，我们研究了所提出的估计器和预测器的渐近性质。 值得注意的是，虽然正则化预测器达到了渐近正态性，但由于自协方差算子的紧致性，自回归算子的相应估计量在算子范数拓扑下未能弱收敛分布。 我们进一步分析了均方预测误差 (MSPE)，将其分解为可归因于正则化偏差、截断和估计方差的部分。 这种分解揭示了我们的方法相对于传统线性截断方案的优势。 广泛的模拟和应用于高频可穿戴传感器数据表明，所提出的方法在捕捉功能时间序列中的复杂时间结构方面具有实际效用和鲁棒性。