A bootstrap procedure for constructing prediction bands 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-Loeve representation of the functional process. It generates backward-in-time, functional replicates that adequately mimic the dependence structure of the underlying process in a model-free way 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 the innovation and estimation errors associated with prediction and the possible errors due to model misspecification. We establish the asymptotic validity of the bootstrap procedure in estimating the conditional prediction error distribution of interest, and we also show that the procedure enables the construction of prediction bands that achieve (asymptotically) the desired coverage. Prediction bands based on a consistent estimation of the conditional distribution of the studentized prediction error process also are introduced. Such bands allow for taking more appropriately into account the local uncertainty of prediction. 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.

翻译：本文提出了一种针对平稳函数型时间序列构建预测带的Bootstrap方法。该方法利用函数过程Karhunen-Loeve展开中傅里叶系数时间反转序列的一般向量自回归表示，通过时间反向生成能够以无模型方式充分模拟底层过程依赖结构的函数型副本，并在每个函数伪时间序列末端保持条件固定的曲线。随后，将无模型Bootstrap生成的未来函数观测值与用于预测的模型所得函数预测值之差作为Bootstrap预测误差分布。这使得估计的预测误差分布能够同时考虑与预测相关的创新误差和估计误差，以及由模型错误设定可能导致的误差。我们证明了该Bootstrap方法在估计目标条件预测误差分布时的渐近有效性，同时表明该方法能够构建实现（渐近）目标覆盖率的预测带。进一步引入了基于学生化预测误差过程条件分布一致估计的预测带，该预测带能够更合理地考虑预测的局部不确定性。通过模拟研究与两个数据集分析，我们验证了所提方法的能力及优异有限样本性能。