Title: Inference for the cross-covariance operator of stationary functional time series Show Chinese title

Title: 平稳函数时间序列的交叉协方差算子的推断

Abstract: When considering two or more time series of functions or curves, for instance those derived from densely observed intraday stock price data of several companies, the empirical cross-covariance operator is of fundamental importance due to its role in functional lagged regression and exploratory data analysis. Despite its relevance, statistical procedures for measuring the significance of such estimators are undeveloped. We present methodology based on a functional central limit theorem for conducting statistical inference for the cross-covariance operator estimated between two stationary, weakly dependent, functional time series. Specifically, we consider testing the null hypothesis that two series possess a specified cross-covariance structure at a given lag. Since this test assumes that the series are jointly stationary, we also develop a change-point detection procedure to validate this assumption, which is of independent interest. The most imposing technical hurdle in implementing the proposed tests involves estimating the spectrum of a high dimensional spectral density operator at frequency zero. We propose a simple dimension reduction procedure based on functional PCA to achieve this, which is shown to perform well in a small simulation study. We illustrate the proposed methodology with an application to densely observed intraday price data of stocks listed on the NYSE. Show Chinese abstract

Abstract: 当考虑两个或多个函数或曲线的时间序列时，例如来自多家公司密集观测的日内股票价格数据，由于其在函数滞后回归和探索性数据分析中的作用，经验交叉协方差算子具有根本的重要性。 尽管它具有相关性，但用于衡量此类估计量显著性的统计程序尚未开发。 我们提出了基于函数中心极限定理的方法来进行两个平稳且弱相关的函数时间序列之间估计的交叉协方差算子的统计推断。 具体而言，我们考虑检验以下零假设：两个序列在给定滞后下具有特定的交叉协方差结构。 由于该检验假设序列是联合平稳的，我们还开发了一种变点检测程序来验证这一假设，这具有独立的研究兴趣。 实现所提出的检验方法的最大技术障碍涉及在频率零点估计高维谱密度算子的谱。 我们提出了一种基于函数主成分分析（PCA）的简单降维程序来实现这一点，并在一个小规模模拟研究中显示其表现良好。 我们通过应用纽约证券交易所上市股票的密集观测日内价格数据来说明所提出的这种方法。