Bootstrap Prediction Bands for Functional Time Series
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A bootstrap procedure for constructing pointwise or simultaneous prediction intervals for a stationary functional time series is proposed. The procedure exploits a general vector autoregressive representation of the time-reversed series of Fourier coefficients appearing in the Karhunen-Loève representation of the functional process. It generates backwards-in-time, functional replicates that adequately mimic the dependence structure of the underlying process and have the same conditionally fixed curves at the end of each functional pseudo-time series. The bootstrap prediction error distribution is then calculated as the difference between the model-free, bootstrap-generated future functional observations and the functional forecasts obtained from the model used for prediction. This allows the estimated prediction error distribution to account for not only the innovation and estimation errors associated with prediction but also the possible errors from model misspecification. We show the asymptotic validity of the bootstrap in estimating the prediction error distribution of interest. Furthermore, the bootstrap procedure allows for the construction of prediction bands that achieve (asymptotically) the desired coverage. These prediction bands are based on a consistent estimation of the distribution of the studentized prediction error process. Through a simulation study and the analysis of two data sets, we demonstrate the capabilities and the good finite-sample performance of the proposed method.
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