函数型数据的统计推断

Statistical inference of functional data is one of the focused research areas of contemporary Mathematical Statistics. Most existing estimation methods for functional data analysis do not come with confidence regions of predetermined confidence levels, and thus can not be used for inference. The main goals of the project are: 1. To estimate the bivariate covariance function of functional data via spline regression and to construct corresponding three dimensional confidence region, and to establish theoretically that its asymptotic confidence level equals what is prescribed; 2. To construct estimator of the number of nonzero eigenvalues in functional data via spline regression and BIC, and investigate its consistency; 3. To estimate the nonzero eigenvalues and the corresponding functional principle components via spleen regression method, to construct confidence intervals and bivariate confidence regions associated with them, and to establish theoretically that their asymptotic confidence levels equal what are prescribed; 4. To construct estimators of the coefficient functions in varying coefficient model functional data via spleen regression together with bivariate confidence regions, and to establish theoretically that their asymptotic confidence levels equal what are prescribed. For each estimator described above, oracle efficiency will be established theoretically.

函数型数据的统计推断是当代数理统计学研究的重点方向之一。多数现存的函数型数据估计方法都不提供预定置信水平的置信区域，也就因此不能用于统计推断。本项目的主要目标是：第一，用样条函数回归方法估计函数型数据的二元协方差函数并构造相应的三维置信区域，并从理论上论证其渐近置信水平等于所要求的；第二，运用类似于BIC的方法和样条函数回归构造函数型数据非零特征值个数的估计量，并研究其一致性；第三，用样条函数回归方法估计函数型数据的非零特征值和相应的函数主分量，构造相应的置信区间和二维置信带，并从理论上论证其渐近置信水平等于所要求的；第四，用样条函数回归方法构造变系数模型函数型数据的系数函数估计量，以及对应的二维置信带，并从理论上论证其渐近置信水平等于所要求的。对于上述每个未知函数的估计量，都从理论上论证其默示有效性。

函数型数据，以及与之密切相关的时间序列，抽样调查, 和复杂高维数据的统计推断都是当代统计学的重点研究方向。本项目 (1) 用样条函数估计稠密函数型数据的二元协方差函数, 用样条函数估计稀疏函数型数据变系数模型的系数函数，分别构造渐近置信水平等于预设理论值的同时置信区域；（2）用核光滑估计自回归时间序列的误差分布函数以及抽样调查数据的总体分布函数，并构造渐近置信水平等于预设理论值的Kolmogorov-Smirnov类同时置信区域；（3）用样条回修核光滑化估计高维数据广义可加模型的分量函数，证明默示有效性并构造了渐近置信水平等于预设理论值的同时置信区域，用核光滑化以及局部线性光滑化估计高维数据单指标模型链接函数，证明其默示有效性并构造了渐近置信水平等于预设理论值的同时置信区域；（4）用样条函数估计ARMA时间序列的趋势以进一步估计ARMA模型系数和阶数并证明了默示有效性，用样条函数估计时间序列半参数GARCH模型并证明了默示有效性；（5）用样条函数和核光滑估计非参数回归模型的条件方差函数和局部相关系数函数，并构造了渐近置信水平等于预设理论值的同时置信区域。