A reproducing Kernel Hilbert space approach to functional linear regression

Annals of Statistics

Volumn 38, Issue 6, 2010, Pages 3412-3444

  Yuan, Ming ^a   Cai, T Tony ^b  

^a Georgia Institute of Technology   (United States)

^b UNIVERSITY OF PENNSYLVANIA   (United States)

Author keywords

Covariance; Eigenfunction; Eigenvalue; Functional linear regression; Minimax; Optimal convergence rate; Principal component analysis; Reproducing kernel Hilbert space; Sacks Ylvisaker conditions; Simultaneous diagonalization; Slope function; Sobolev space

Indexed keywords

EID: 78650085539     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOS772     Document Type: Article

References (24)

1. ADAMS, R. A. (1975). Sobolev Spaces. Academic Press, New York. MR0450957
2. CAI, T. and HALL, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159-2179. MR2291496
3. CARDOT, H., FERRATY, F. and SARDA, P. (2003). Spline estimators for the functional linear model. Statist. Sinica 13 571-591. MR1997162 (Pubitemid 37155842)
4. CARDOT, H. and SARDA, P. (2005). Estimation in generalized linear models for functional data via penalized likelihood. J. Multivariate Anal. 92 24-41. MR2102242
5. COX, D. D. and O'SULLIVAN, F. (1990). Asymptotic analysis of penalized likelihood and related estimators. Ann. Statist. 18 1676-1695. MR1074429
6. CRAMBES, C., KNEIP, A. and SARDA, P. (2009). Smoothing splines estimators for functional linear regression. Ann. Statist. 37 35-72. MR2488344
7. CUCKER, F. and SMALE, S. (2001). On the mathematical foundations of learning. Bull. Amer. Math. Soc. 39 1-49. MR1864085
8. FERRATY, F. and VIEU, P. (2006). Nonparametric Functional Data Analysis: Methods, Theory, Applications and Implementations. Springer, New York. MR2229687
9. HALL, P. and HOROWITZ, J. L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35 70-91. MR2332269
10. JAMES, G. (2002). Generalized linear models with functional predictors. J. Roy. Statist. Soc. Ser. B 64 411-432. MR1924298
11. JOHANNES, J. (2009). Nonparametric estimation in functional linear models with second order stationary regressors. Unpublished manuscript. Available at http://arxiv.org/abs/0901.4266v1.
12. LI, Y. and HSING, T. (2007). On the rates of convergence in functional linear regression. J. Multivariate Anal. 98 1782-1804. MR2392433
13. MICCHELLI, C. and WAHBA, G. (1981). Design problems for optimal surface interpolation. In Approximation Theory and Applications (Z. Ziegler, ed.) 329-347. Academic Press, New York. MR0615422
14. MÜLLER, H. G. and STADTMÜLLER, U. (2005). Generalized functional linear models. Ann. Statist. 33 774-805. MR2163159
15. RAMSAY, J. O. and SILVERMAN, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York. MR2168993
16. RIESZ, F. and SZ-NAGY, B. (1955). Functional Analysis. Ungar, New York. MR0071727
17. RITTER, K., WASILKOWSKI, G. and WOŹNIAKOWSKI, H. (1995). Multivariate integeration and approximation for random fields satisfying Sacks-Ylvisaker conditions. Ann. Appl. Probab. 5 518-540. MR1336881
18. SACKS, J. and YLVISAKER, D. (1966). Designs for regression problems with correlated errors. Ann. Math. Statist. 37 66-89. MR0192601
19. SACKS, J. and YLVISAKER, D. (1968). Designs for regression problems with correlated errors; many parameters. Ann. Math. Statist. 39 49-69. MR0220424
20. SACKS, J. and YLVISAKER, D. (1970). Designs for regression problems with correlated errors III. Ann. Math. Statist. 41 2057-2074. MR0270530
21. SILVERMAN, B. W. (1982). On the estimation of a probability density function by the maximum penalized likelihood method. Ann. Statist. 10 795-810. MR0663433
22. STEIN, M. (1999). Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York. MR1697409
23. WAHBA, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia. MR1045442
24. YAO, F., MÜLLER, H. G. and WANG, J. L. (2005). Functional linear regression analysis for longitudinal data. Ann. Statist. 33 2873-2903. MR2253106 (Pubitemid 43384879)