Yule-Walker Estimation for Functional Time Series in Hilbert Space (2506.05922v1)

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.