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：本文提出一种用于构建平稳函数型时间序列预测带的Bootstrap方法。该方法利用函数过程Karhunen-Loeve展开中傅里叶系数的时间反转序列所对应的一般向量自回归表示，通过反向时间生成充分模拟基础过程依赖结构（无模型依赖）的函数复制序列，并使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，从而使得估计的预测误差分布能够兼顾与预测相关的创新误差、估计误差以及可能由模型设定不当导致的误差。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，并展示了该方法能够构建达到（渐近）期望覆盖率的预测带。同时引入基于学生化预测误差过程条件分布一致估计的预测带，此类预测带能更恰当地考虑局部预测不确定性。通过仿真实验和两个数据集分析，验证了所提方法的表现与优良的有限样本性能。