Convolutional autoregressive models for functional time series

Functional data analysis has became an increasingly popular class of problems in statistical research. However, functional data observed over time with serial dependence remains a less studied area. Motivated by Bosq (2000), who first introduced the functional autoregressive models, we propose a convolutional functional autoregressive model, where the function at time t is a result of the sum of convolutions of the past functions and a set of convolution functions, plus a noise process, mimicking the vector autoregressive process. It provides an intuitive and direct interpretation of the dynamics of a stochastic process. Instead of principal component analysis commonly used in functional data analysis, we adopt a sieve estimation procedure based on B-spline approximation of the convolution functions. We establish convergence rate of the proposed estimator, and investigate its theoretical properties. The model building, model validation, and prediction procedures are also developed. Both simulated and real data examples are presented.