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。

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.