Local asymptotics for regression splines and confidence regions

Annals of Statistics

Volumn 26, Issue 5, 1998, Pages 1760-1782

  Zhou, S ^a   Shen, X ^a   Wolfe, D A ^a  

^a OHIO STATE UNIVERSITY   (United States)

Author keywords

Asymptotic bias; Asymptotic normality; Asymptotic variance; Bernoulli polynomial; Confidence interval and confidence band; Least squares; Regression splines

Indexed keywords

EID: 0032263988     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1024691356     Document Type: Article

References (26)

1. AGARWAL, G. G. and STUDDEN W. J. (1980). Asymptotic integrated mean square error using least squares and bias minimizing splines. Ann. Statist. 8 1307-1325.
2. 2[0,1] approximation by spline with variable knots. Quart. Appl. Math. 36 293-304.
3. CRAVEN, P. and WAHBA, G. (1979). Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the generalized cross-validation. Numer. Math. 31 337-403.
4. DE BOOR, C. (1972). On calculating with B-splines. J. Approx. Theory 6 50-62.
5. DE BOOR, C. (1978). A Practical Guide to Splines. Springer, New York.
6. DEMKO, S. (1977). Inverses of band matrices and local convergence of spline projection. SIAM J. Numer. Anal. 14 616-619.
7. FRIEDMAN, J. H. (1991). Multivariate adaptive regression splines (with discussion). Ann. Statist. 19 1-141.
8. GAENSSLER, P. and WELLNER, J. A. (1981). Glivenko-Cantelli theorems. Encyclopedia of Statistical Science 3 442-445. Wiley, New York.
9. GASSER, T. and MÜLLER, H. G. (1984). Estimating regression functions and their derivatives by the kernel method. Scand. J. Statist. 11 171-185.
10. GASSER, T., SROKA, L. and JENNEN-STEINMETZ, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73 625-633.
11. HALL, P., KAY, J. W. and TITTERINGTON, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521-528.
12. HUANG, S. Y. and STUDDEN, W. Y. (1993). An equivalent kernel method for least squares spline regression. Statist. Decisions 3 179-201 (Supplement).
13. JOHANSEN, S. and JOHNSTONE, M. (1990). Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis. Ann. Statist. 18 652-684.
14. KOENKER, R., NG, P. and PORTNOY, S. (1994). Non-parametric estimation of conditional quantile functions. Biometrika 81 673-680.
15. 2 rate of convergence for hazard regression. Scand. J. Statist. 22 143-157.
16. NADARAYA, E. A. (1964). On estimating regression. Theory Probab. Appl. 9 141-142.
17. PORTNOY, S. (1997). Local asymptotics for quantile smoothing splines. Ann. Statist. 25 414-434.
18. SCHUMAKER, L. L. (1981). Spline Functions: Basic Theory. Wiley, New York.
19. SHEN, X. and HU, C. (1995). Universal sieves and spline adaptation. Technical Report 543, Dept. Statist., Ohio State Univ.
20. 1-norm" estimates in a partly linear model. Comm. Statist. Theory Methods 23 175-196.
21. SHI, P. and LI, G. (1995). Global convergence rates of B-spline M-estimators in nonparametric regression. Statist. Sinica 5 303-318.
22. STONE, C. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040-1053.
23. STONE, C. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689-705.
24. STONE, C. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590-606.
25. STONE, C. (1994). The use of polynomial splines and their tensor products in multivariate function estimation. Ann. Statist. 22 118-184.
26. STONE, C. and KOO, C. Y. (1986). Additive splines in statistics. In Proceedings of the Statistical Computing Section 45-48. Amer. Statist. Assoc., Washington, DC.