Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions

Annals of Statistics

Volumn 35, Issue 6, 2007, Pages 2589-2619

  Horowitz, Joel L ^a   Mammen, Enno ^b  

^a NORTHWESTERN UNIVERSITY   (United States)

^b UNIVERSITY OF MANNHEIM   (Germany)

Author keywords

Empirical process methods; Generalized additive models; Multivariate curve estimation; Nonparametric regression; Penalized least squares; Smoothing splines

Indexed keywords

EID: 50849145175     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053607000000415     Document Type: Article

References (43)

1. ADAMS, R. A. (1975). Sobolev Spaces. Academic Press, New York. MR0450957
2. AGMON, S. (1965). Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton, NJ. MR0178246
3. p. Mat. Sb. (N. S.) 73 295-317. MR0217487
4. BREIMAN, L. and FRIEDMAN, J. H. (1985). Estimating optimal transformations for multiple regression and correlation (with discussion). J. Amer. Statist. Assoc. 80 580-619. MR0803258
5. BUJA, A., HASTIE, T. and TIBSHIRANI, R. (1989). Linear smoothers and additive models (with discussion). Ann. Statist. 17 453-555. MR0994249
6. CHRISTOPEIT, N. and HODERLEIN, S. (2006). Local partitioned regression. Econometrica 74 787-817. MR2217617
7. COPPEJANS, M. (2004). On Kolmogorovs representation of functions of several variables by functions of one variable. J. Econometrics 123 1-31. MR2125436
8. EUBANK, R. L. (1988). Spline Smoothing and Nonparametric Regression. Dekker, New York. MR0934016
9. FAN, J., HÄRDLE, W. and MAMMEN, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26 943-971. MR1635422
10. HÄRDLE, W., HUET, S., MAMMEN, E. and SPERLICH, S. (2004). Bootstrap inference in semiparametric generalized additive models. Econometric Theory 20 265-300. MR2044272
11. HASTIE, T. and TIBSHIRANI, R. (1990). Generalized Additive Models. Chapman and Hall, London. MR1082147
12. HOROWITZ, J. (2001). Nonparametric estimation of a generalized additive model with an unknown link function. Econometrica 69 499-513. MR1819761
13. HOROWITZ, J., KLEMELÄ, J. and MAMMEN, E. (2006). Optimal estimation in additive regression models. Bernoulli 12 271-298. MR2218556
14. HOROWITZ, J. and MAMMEN, E. (2004). Nonparametric estimation of an additive model with a link function. Ann. Statist. 32 2412-2443. MR2153990
15. HOROWITZ, J. and MAMMEN, E. (2006). Nonparametric estimation of a generalized additive model with an unknown link function. Preprint.
16. ICHIMURA, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71-120. MR1230981
17. JUDITSKY, A. B., LEPSKI, O. V. and TSYBAKOV, A. B. (2007). Nonparametric estimation of composite functions. Preprint.
18. KAUERMANN, G. and OPSOMER, J. D. (2003). Local likelihood estimation in generalized additive models. Scand. J. Statist. 30 317-337. MR1983128
19. LINTON, O. (2000). Efficient estimation of generalized additive nonparametric regression models. Econometric Theory 16 502-523. MR1790289
20. LINTON, O. and HÄRDLE, W. (1996). Estimation of additive regression models with known links. Biometrika 83 529-540. MR1423873
21. LINTON, O. and NIELSEN, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93-100. MR1332841
22. MAMMEN, E., LINTON, O. and NIELSEN, J. P. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443-1490. MR1742496
23. MAMMEN, E. and NIELSEN, J. P. (2003). Generalised structured models. Biometrika 90 551-566. MR2006834
24. MAMMEN, E. and PARK, B. U. (2005). Bandwidth selection for smooth backfitting in additive models. Ann. Statist. 33 1260-1294. MR2195635
25. MAMMEN, E. and PARK, B. U. (2006). A simple smooth backfitting method for additive models. Ann. Statist. 34 2252-2271. MR2291499
26. MAMMEN, E. and THOMAS-AGNAN, C. (1999). Smoothing splines and shape restrictions. Scand. J. Statist. 26 239-252. MR1707587
27. MAMMEN, E. and VAN DE GEER, S. (1997). Penalized quasi-likelihood estimation in partial linear models. Ann. Statist. 25 1014-1035. MR1447739
28. MCCULLAGH, P. and NELDER, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman and Hall, London.
29. NELDER, J. A. and WEDDERBURN, R. W. M. (1972). Generalized linear models. J. Roy. Statist. Soc. Ser. A 135 370-384.
30. NEWEY, W. K. (1997). Convergence rates and asymptotic normality for series estimators. J. Econometrics 79 147-168. MR1457700
31. NIELSEN, J. P. and SPERLICH, S. (2005). Smooth backfitting in practice. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 43-61. MR2136638
32. ODEN, J. T. and REDDY, J. N. (1976). An Introduction to the Mathematical Theory of Finite Elements. Wiley, New York. MR0461950
33. OPSOMER, J. D. (2000). Asymptotic properties of backfitting estimators. J. Multivariate Anal. 73 166-179. MR1763322
34. OPSOMER, J. D. and RUPPERT, D. (1997). Fitting a bivariate additive model by local polynomial regression. Ann. Statist. 25 186-211. MR1429922
35. STONE, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689-705. MR0790566
36. STONE, C. J. (1986). The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590-606. MR0840516
37. STONE, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation (with discussion). Ann. Statist. 22 118-184. MR1272079
38. TJØSTHEIM, D. and AUESTAD, B. H. (1994). Nonparametric identification of nonlinear time series: Projections. J. Amer. Statist. Assoc. 89 1398-1409. MR1310230
39. VAN DE GEER, S. (1990). Estimating a regression function. Ann. Statist. 18 907-924. MR1056343
40. VAN DE GEER, S. (2000). Empirical Processes in M-Estimation. Cambridge Univ. Press.
41. WEDDERBURN, R. W. M. (1974). Quasi-likelihood functions, generalized linear models and the Gauss-Newton method. Biometrika 61 439-447. MR0375592
42. YOSIDA, K. (1974). Functional Analysis, 4th ed. Springer, New York. MR0350358
43. YU, K., PARK, B. U. and MAMMEN, E. (2007). Smooth backfitting in generalized additive models. Ann. Statist. To appear.