复杂时间序列与函数型数据的统计分析

Stochastic process data present great challenges to statisticians, due to their complex dependence structure. This project aims at developing tools of statistical inference for nonstationary financial time series data and for dense functional data with nuisance parameters, as well as for censored and truncated dense functional data. Specifically, the project goals are: (1) to obtain oracally efficient two-step estimators for the ARCH/GARCH parameters of a time varying ARCH/GARCH model based on residuals by removing a slowly varying nonparametric trend; (2) to formulate smooth kernel estimator of the distribution of the hidden stationary ARCH/GARCH process based on residuals from a subsample, that is near oracally efficient with asymptotic simultaneous confidence band; (3) to formulate oracally efficient spline and local polynomial estimators with asymptotic simultaneous confidence band for the population mean function of dense function data with holiday effects, and obtain oracally efficient estimators for the holiday effects parameters; (4) to formulate oracally efficient spline and local polynomial estimators with asymptotic simultaneous confidence band for the population mean function of censored and truncated dense function data; and (5) to obtain oracally efficient spline estimators with asymptotic simultaneous confidence band of the bivariate covariance functions and univariate variance functions. The theoretical results will be applied to analyze financial big data such as S&P 500 daily returns which exhibit non stationarity over long horizon, for risk management and forecasting， and to employee learning curve data such as sports store employee sales performance.

随机过程数据以其复杂相依结构，对统计学提出巨大的挑战。本项目对于非平稳金融时间序列数据, 和带有冗余参数, 以及被删失和截断的稠密函数型数据建立统计推断工具。具体目标是：（1）基于去除缓变非参数趋势的残差，得到时变ARCH/GARCH模型的ARCH/GARCH参数的默示有效两步估计；（2）对隐平稳ARCH/GARCH时间序列的分布函数构造光滑核估计，基于子样本的残差，具有近似默示有效的渐近同时置信带；（3）对带有节假日效应的稠密函数型数据的总体均值函数构造默示有效样条与局部多项式估计，以及渐近同时置信带；（4）对被删失和截断的稠密函数型数据的总体均值函数构造默示有效样条与局部多项式估计，以及渐近同时置信带；（5）对二元协方差函数和一元方差函数，获得默示有效样条估计以及渐近同时置信带。理论结果将用于长期非平稳金融大数据如S&P500的风险管理与预测，以及体育用品店员工学习曲线数据的分析。

随机过程数据以其复杂相依结构，对统计学提出巨大的挑战。本项目对于复杂时间序列数据与函数型数据建立了一系列统计推断工具。包括：（1） 自回归时间序列的默示有效多步向前预测区间构造 （2）时变ARCH模型的默示有效两步估计构造 （3）带有节假日效应的稠密函数型数据的均值函数默示有效估计以及渐近同时置信带构造（4）二元协方差函数和一元方差函数的默示有效估计以及渐近同时置信带构造（5）函数型数据误差分布函数的渐近同时置信带构造（6）脑电信号对工作记忆与辩证思维等神经科学指标的预测模型构造（7）由数值预测图像回归模型的统计推断（8）非参数回归误差分布函数的渐近同时置信带构造（9）两个非参数回归方差函数之比的渐近同时置信带构造（10）广义空间可加模型的统计推断（11）可加前沿函数的估计（12）分层抽样调查数据总体分布函数的渐近同时置信带构造（13）时间序列分布函数的渐近同时置信带构造（14）函数型时间序列均值函数的默示有效估计与渐近同时置信带构造（15）非参数回归分析空气污染物浓度的预测区间（16）两个非参数回归均值函数之差的渐近同时置信带构造。

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