函数型时间序列的频域自助法
Frequency Domain Bootstrap for Functional Time Series
AI总结：
本文提出用于函数型时间序列的频域自助法，通过相关重采样程序实现对谱平均算子二阶及四阶结构的一致估计，在弱条件下验证了该方法的有效性。
AI中文摘要：
本文提出一种针对函数型时间序列的频域自助法程序，并将其应用于谱平均算子类。该程序的运作方式为：首先利用涉及的谱密度算子估计量，在正傅里叶频率上生成独立的伪周期图算子；随后生成所关注谱平均算子的函数型重复样本。通过对自助后的谱平均算子进行基于投影的加性分解，其主导的m维部分得到恰当补充，以捕获过程的相关四阶特征，该补充通过基于子样本卷积周期图算子的重采样程序实现。所得自助谱平均算子可一致估计谱平均算子分布的全部二阶结构及m维四阶结构。通过允许分解参数m随样本量趋于无穷而趋于无穷，还可实现对过程全部四阶结构估计的一致性。所建立的渐近理论研究了固定m及递增m时该程序的性质，并在潜在函数型过程类的相当弱条件下，确立了该频域自助法方案的有效性。
英文摘要：
A frequency domain bootstrap procedure for functional time series is proposed and applied to the class of spectral mean operators. The procedure works by first generating independent pseudo periodogram operators across the positive Fourier frequencies using an estimator of the spectral density operator involved. Functional replicates of the spectral mean operators of interest are then generated. Through an additive, projection-based decomposition of the bootstrapped spectral mean operator, its leading $m$-dimensional part is properly complemented to also capture the relevant fourth order characteristics of the process. The complementation is achieved by means of a resampling procedure based on convolved periodogram operators of subsamples. The resulting bootstrap spectral mean operator consistently estimates the entire second order as well as the $m$-dimensional fourth order structure of the distribution of spectral mean operators. By allowing for the decomposition parameter $m$ to increase to infinity as the sample size increases to infinity, consistency in estimating the entire fourth order structure of the process also is achieved. The asymptotic theory developed investigates properties of the procedure for fixed and for increasing $m$ and establishes validity of the frequency domain bootstrap proposal under rather weak conditions on the underlying functional process class.
