Curve estimation when the design density is low

Annals of Statistics

Volumn 25, Issue 2, 1997, Pages 756-770

  Hall, Peter ^a   Marron, J S ^b   Neumann, M H ^c   Titterington, D M ^d  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^c The University of North Carolina at Chapel Hill   (United States)

^d UNIVERSITY OF GLASGOW   (United Kingdom)

Author keywords

Bandwidth choice; Design density; Kernel methods; Local adaptivity; Local linear smoothing; Projection pursuit; Single index model

Indexed keywords

EID: 0031520798     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1031833672     Document Type: Article

References (27)

1. BRILLINGER, D. R. (1983). A generalized linear model with "Gaussian" regressor variables. In A Festschrift for Erich Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, eds.) 97-114. Wadsworth, Belmont, CA.
2. BUCKLEY, M. J. and EAGLESON, G. K. (1989). A graphical method for estimating the residual variance in nonparametric regression. Biometrika 76 203-210.
3. BUCKLEY, M. J., EAGLESON, G. K. and SILVERMAN, B. W. (1988). The estimation of residual variance in nonparametric regression. Biometrika 75 189-199.
4. CARTER, C. K. and EAGLESON, G. K. (1992). A comparison of variance estimators in nonparametric regression. J. Roy. Statist. Soc. Sen B 54 773-780.
5. CHENG, M. Y., FAN, J. and MARRON, J. S. (1996). Minimax efficiency of local polynomial fit estimators at boundaries. Mimeo Series No. 2098, Institute of Statistics, Univ. North Carolina.
6. CLEVELAND, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74 829-836.
7. CLEVELAND, W. S. (1993). Visualizing Data. Hobart Press, Summit, NJ.
8. CLEVELAND, W. S. and DEVLIN, S. J. (1988). Locally weighted regression: an approach to regression analysis by local fitting. J. Amer. Statist. Assoc. 83 596-610.
9. CLEVELAND, W. S. and GROSSE, E. H. (1991). Computational methods for local regression. Statist. and Computing 1 47-62.
10. CLEVELAND, W. S. and LOADER, C. (1996). Smoothing by local regression: principles and methods. In Statistical Theory and Computational Aspects of Smoothing (W. Härdle and M. Schmlick, eds.) 10-49. Physica, Heidelberg.
11. FAN, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 998-1004.
12. FAN, J. (1993). Local linear smoothers and their minimax efficiency. Ann. Statist. 21 196-216.
13. FAN, J. and GIJBELS, I. (1995). Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation. J. Roy. Statist. Soc. Ser. B 57 371-394.
14. FRIEDMAN, J. H. and STUETZLE, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817-823.
15. GASSER, T., KNEIP. A. and KÖHLER, W. (1991). A flexible and fast method for automatic smoothing. J. Amer. Statist. Assoc. 86 643-652.
16. GASSER, T., SROKA, L. and JENNEN-STEINMETZ, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73 625-633.
17. HALL, P. and HEYDE, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
18. HALL, P., KAY, J. W. and TITTERINGTON, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521-528.
19. HALL, P. and MARRON, J. S. (1990). On variance estimation in nonparametric regression. Biometrika 77 415-419.
20. HALL, P., MARRON, J. S., NEUMANN, M. H. and TITTERINGTON, D. M. (1995). On local linear smoothing when the design density is low. Research Report SRR027-95, Centre for Mathematics and Its Applications, Australian National Univ., Canberra.
21. HARDLE, W., HALL, P. and ICHIMURA, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157-178.
22. HASTIE, T. and LOADER, C. (1993). Local regression: automatic kernel carpentry. Statist. Sci. 8 120-143.
23. NICHOLLS, D. F. and PAGAN, A. R. (1985). Varying coefficient regression. In Handbook of Statistics 5 (E. J. Hannan, P. R. Krishnaiah and M. M. Rao, eds.) 413-449. North-Holland, Amsterdam.
24. RAJ, B. and ULLAH, A. (1981). Econometrics: A Varying Coefficients Approach. Croom-Helm, London.
25. RUPPERT, D. and WAND, M. P. (1994). Multivariate locally weighted least squares regression. Ann. Statist. 22 1346-1370.
26. SEIFERT, B., GASSER, T. and WOLF, A. (1993). Nonparametric estimation of residual variance revisited. Biometrika 80 373-383.
27. SEIFERT, B. and GASSER, T. (1996). Finite sample analysis of local polynomials: analysis and solutions. In Statistical Theory and Computational Aspects of Smoothing (W. Härdle and M. Schmlick, eds.) 50-79. Physica, Heidelberg.