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：本文提出了一种构建平稳函数型时间序列预测带的Bootstrap方法。该方法利用了函数过程Karhunen-Loeve展开中傅里叶系数的时间反转序列的一般向量自回归表示，通过逆向时间生成函数化复制——这些复制以无模型方式充分模仿原始过程的相依结构，并在每个函数化伪时间序列末端保持相同的条件固定曲线。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到的函数预测值之间的差异，定义为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预测相关的创新误差与估计误差，以及模型设定错误可能导致的误差。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，并表明该方法能够构建（渐近地）实现预期覆盖率的预测带。本文还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，此类预测带能更恰当地考虑预测的局部不确定性。通过模拟研究和两个数据集的分析，我们展示了所提出方法的性能与良好的有限样本表现。