Abstract
This paper considers a nonparametric functional autoregression model of order one. Existing contributions addressing the problem of functional time series prediction have focused on the linear model and literatures are rather lacking in the context of nonlinear functional time series. In our nonparametric setting, we define the functional version of kernel estimator for the autoregressive operator and develop its asymptotic theory under the assumption of a strong mixing condition on the sample. The results are general in the sense that high-order autoregression can be naturally written as a first-order AR model. In addition, a component-wise bootstrap procedure is proposed that can be used for estimating the distribution of the kernel estimation and its asymptotic validity is theoretically justified. The bootstrap procedure is implemented to construct prediction regions that achieve good coverage rate. A supporting simulation study is presented in the end to illustrate the theoretical advances in the paper.
Highlights
Popularized by the pioneering works of Ramsay and Silverman (1997) [29], (2002) [30], Functional Data Analysis (FDA) has emerged as a promising field of statistical research in the past decade
When functional data objects being collected sequentially over time that exhibit forms of dependence, such data are known as functional time series
The typical situation in which functional time series arise is when long continuous records of temporal sequence are segmented into curves over natural consecutive time intervals
Summary
Popularized by the pioneering works of Ramsay and Silverman (1997) [29], (2002) [30], Functional Data Analysis (FDA) has emerged as a promising field of statistical research in the past decade. Pan and Politis (2016) [25] developed a coherent methodology for the construction of bootstrap prediction intervals, which can be successfully applied to the nonlinear univariate autoregression models Those results can be naturally extended to multivariate time series, but that is not the case for functional time series due to the infinite dimensional nature of functional data. While it is more of the interest to study the estimator Ψh(χ), the need for model (2.4) and the estimator ψh(χ) will be seen in Section 5 where a componentwise bootstrap approximation is proposed
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