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中心概述

当今数理统计学的主流是复杂数据的统计推断,如高维数据,多元时间序列数据,等等。自2000 年以来欧美统计界开始关注更为复杂的物体型数据分析(Analysis of Object Data缩写为AOD), 例如2010-2011 年在美国北卡罗来纳州的统计与应用数学科学研究院(Statistical and Applied Mathematical Sciences Institute 缩写为SAMSI)就举办了由美国国家科学基金(NSF)资助的一整年的AOD 大型科研项目,包括一系列的Workshops, 专题课,由来自加州大学戴维斯分校,北卡罗来纳大学,普林斯顿大学等顶级统计系的著名教授主持并讲课,有大批博士生与博士后参与。

正如SAMSI 在其网站上面对该AOD 科研项目先容的(http://www.samsi.info/programs/2010-11-program-analysis-object-data),AOD 的应用范围涉及高维图像分析,生物信息数据分析,进化生物学,电子商务,心理学与社会科学等当今的尖端科技。其中需要的数学工具也涵盖了概率论与数理统计,微分几何,拓扑,微分方程,优化计算方法等学科。正因为这样的交叉学科性质,物体型数据分析方法的研究代表了当代统计学的最高水平。相对于其它物体型数据,函数型数据(又称曲线数据)的统计理论是较为成熟的。其中以F. Ferraty和P. Vieu为代表的法国学派(见书[4])研究的对象是n个独立同分布,连续时间观测的随机过程,而以J. O. Ramsay和B.W. Silverman为代表的英美学派研究的对象,则是更符合实际数据特性的离散时间观测,且含有观测误差的纵向数据所包含的函数结构(见书[14])。近年来函数型数据的研究成果主要来自英美派学者,除了前面提到的Ramsay和Silverman,以下英美学派的知名统计学家也发表了很多关于函数型数据的有影响的论文:R. J Carroll, P. Hall, T. J. Hastie, N. Heckman, T. Hsing, G. M. James, J.S. Marron, J. S. Morris, H. G. Müller, C. A. Sugar, J. L. Wang, L. Yang, F. Yao,见参考文献[1-17]。国内的知名统计学家耿直,林华珍,孙六权,朱仲义等,近年来在与函数型数据相关的纵向数据方面的研究取得了许多引起国际重视的成果,见参考文献[18-21]。

函数型数据不是简单的多元数据,尽管形式上十分类似。因为同一条曲线上按不同时间观测到的点是按顺序出现的,且有拓扑结构,不可以任意排列。此外,函数型数据的协方差函数是二维连续区域上的函数,不对应于多元数据的固定维数的协方差矩阵。 这些使得函数型数据的统计理论较一般高维数据复杂得多。

在非参数统计推断中占据非常重要地位的同时置信带,是直到最近才由杨立坚等人对函数型数据提出并给出严格的数学论证,见Cao, Wang, Wang and Todem (2012)[1], Cao, Yang and Todem (2012)[2], Degras (2011)[3], Ma, Yang and Carroll (2012)[11]。除了上述参考文献 [2,11],申请人近3年来还发表了一系列关于非参数同时置信带的论文,如Wang and Yang (2010) [26], Song and Yang (2010) [27], Song and Yang (2009) [29], Wang and Yang (2009)[30]。国内的一些知名统计学家如周勇等,也在非参数同时置信带方向做出过重要的贡献,见参考文献[22]。本项目主要研究函数型数据的协方差函数的基本性质,包括二元协方差函数的估计量及渐近置信区域,非零特征值个数的估计量及其渐近一致性,非零特征值及其相应的特征函数(又称函数主分量)的估计量及相应的渐近置信带;变系数模型函数型数据的系数函数的估计量,以及对应的渐近置信带。这些问题都是函数型数据长期未解决的难题,也是当前主流统计学所关心的问题。

参考文献:

1. Cao, G., Wang, J., Wang, L. and Todem, D. (2012) Spline confidence bands for functional derivatives. Journal of Statistical Planning and Inference 142, 1557-1570.

2. Cao, G., Yang, L. and Todem, D. (2012) Simultaneous inference for the mean function based on dense functional data. Journal of Nonparametric Statistics 24,359-377.

3. Degras, D. (2011) Simultaneous confidence bands for nonparametric regression with functional data. Statistica Sinica 21, 1735-1765.

4. Ferraty, F. and Vieu, P. (2006) Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics, Springer: Berlin.

5. Hall, P. and Heckman, N. (2002) Estimating and depicting the structure of a distribution of random functions. Biometrika 89, 145-158.

6. Izem, R. and Marron, J. S. (2007) Analysis of nonlinear modes of variation for functional data. Electronic Journal of Statistics 1, 641-676.

7. James, G. M., Hastie, T. J. and Sugar, C. A. (2000) Principal component models for sparse functional data. Biometrika 87,587-602.

8. James, G. M. and Silverman, B. W. (2005) Functional adaptive model estimation. Journal of the American Statistical Association 100,565-576.

9. Li, Y. and Hsing, T. (2007) On rates of convergence in functional linear regression. Journal of Multivariate Analysis 98, 1782-1804.

10. Li, Y. and Hsing, T. (2010) Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Annals of Statistics 38, 3321-3351.

11. Ma, S., Yang, L. and Carroll, R. (2012) A simultaneous confidence band for sparse longitudinal regression. Statistica Sinica 22, 95-122.

12. Morris, J. S. and Carroll, R. J. (2006) Wavelet-based functional mixed models. Journal of the Royal Statistical Society, Series B 68,179-199.

13. Müller, H. G., and Yao, F. (2008) Functional additive models. Journal of American Statistical Association 103, 1534-1544.

14. Ramsay, J. O. and Silverman, B. W. (2005) Functional Data Analysis. Second Edition. Springer Series in Statistics. Springer: New York.

15. Yao, F., Müller, H. G. and Wang, J. L. (2005) Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association 100, 577-590.

16. Yao, F. (2007) Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data. Journal of Multivariate Analysis 98, 40-56.

17. Zhou, L., Huang, J. and Carroll, R. J. (2008) Joint modelling of paired sparse functional data using principal components. Biometrika 95, 601-619.

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19. Lin, H., Song, Peter X.-K. and Zhou, Q. M. (2007) Varying-coefficient marginal models and applications in longitudinal data analysis. Sankhyā 69, 581-614.

20. Sun, J., Sun, L. and Liu, D. (2007) Regression analysis of longitudinal data in the presence of informative observation and censoring times. J. Amer. Statist. Assoc. 102, 1397-1406. 21. Zhu, Z., Fung, W. K. and He, X. (2008) On the asymptotics of marginal regression splines with longitudinal data. Biometrika 95, 907-917.

22. Zhou, Y. and Sun, L. (2005) Sequential confidence bands for quantile densities under truncated and censored data. Acta Math. Appl. Sin. Engl. Ser. 21, 311-322.

23. Wang, L., Li, H. and Huang, J. (2008) Variable selection in nonparametric varying-coefficient models for analysis of repeated measurements. Journal of the American Statistical Association 103, 1556-1569.

24. Kato, T. (1976) Perturbation theory for linear operators. Second edition. Springer-Verlag, Berlin-New York. 1101.02.009.128

25. Piterbarg, Vladimir I. (1996) Asymptotic methods in the theory of Gaussian processes and fields. American Mathematical Society, Providence, RI.

26. Wang, L. and Yang, L. (2010) Simultaneous confidence bands for time series prediction function. Journal of Nonparametric Statistics 22, 999-1018.

27. Song, Q. and Yang, L. (2010) Oracally efficient spline smoothing of nonlinear additive autoregression model with simultaneous confidence band. Journal of Multivariate Analysis 101, 2008-2025.

28. Liu, R. and Yang, L. (2010) Spline-backfitted kernel smoothing of additive coefficient model. Econometric Theory 26, 29-59.

29. Song, Q. and Yang, L. (2009) Spline confidence bands for variance function. Journal of Nonparametric Statistics 21, 589-609.

30. Wang, J. and Yang, L. (2009) Polynomial spline confidence bands for regression curves. Statistica Sinica 19, 325-342.

31. Wang, L. and Yang, L. (2007) Spline-backfitted kernel smoothing of nonlinear additive autoregression model. Annals of Statistics 35, 2474-2503.

32. Xue, L. and Yang, L. (2006) Additive coefficient modeling via polynomial spline. Statistica Sinica 16, 1423-1446.

33. Huang, J. and Yang, L. (2004) Identification of nonlinear additive autoregressive models. Journal of the Royal Statistical Society Series B 66, 463-477.

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