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。

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.