A new class of functional conditional autoregressive models

2026-05-21

AI Summary

This study addresses the challenge of modeling and inference for functional data exhibiting spatial dependence by proposing a class of functional conditional autoregressive models. The dependence structure is characterized through the conditional mean and covariance operators, along with a spatial dependence parameter, all defined with respect to neighboring functional observations. A three-stage estimation procedure is developed, which innovatively yields a super-consistent and asymptotically normal estimator for the spatial dependence parameter—enabling valid statistical inference for the first time in this context. Furthermore, a conditionally centered estimator for the covariance operator is introduced to overcome the inconsistency inherent in conventional marginally centered approaches. Theoretical results establish the consistency and asymptotic properties of the proposed estimators, while numerical experiments demonstrate computational efficiency. The method is successfully applied to analyze weekly PM2.5 concentration trajectories across the Midwestern United States.

Abstract

We introduce a new class of conditional autoregressive models for spatially dependent functional data, formulated through conditional means given neighboring functional observations and characterized by a covariance operator and a spatial dependence parameter. Our estimation strategy consists of three components: (i) estimating the covariance operator using conditionally centered data, (ii) estimating the spatial dependence parameter by maximizing the likelihood of projected observations, and (iii) applying a novel profile-based approach to obtain the final estimators. Under an expanding lattice framework, we establish two key theoretical results. First, we establish the consistency of the proposed covariance estimator, which is not attainable using naive methods based on marginally centered data. Second, we prove that the spatial dependence parameter estimator is superconsistent and asymptotically normal, where the latter property enables statistical inference for spatial dependence in functional data -- a contribution that is novel in the existing literature. Numerical studies support the theoretical results and demonstrate the computational efficiency of our method. Finally, we illustrate its practical utility by analyzing weekly PM$_{2.5}$ concentration trajectories in 2019 across counties in the Midwestern United States.

Problem

• functional data
• spatial dependence
• conditional autoregressive models
• covariance operator
• statistical inference

Innovation

• functional data
• conditional autoregressive model
• spatial dependence
• covariance operator
• superconsistency