Rates of convergence of estimates, Kolmogorov's entropy and the dimensionality reduction principle in regression

Annals of Statistics

Volumn 25, Issue 6, 1997, Pages 2493-2511

  Nicoleris, Theodoros ^a   Yatracos, Yannis G ^b  

^a Université de Montréal ^*  (Greece)

^b UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

Additive and multiplicative regression; Dimensionality reduction principle; Hoeffding's inequality; Interactions; Kolmogorov's entropy; Model selection; Nonparametric regression; Optimal rates of convergence; Projection pursuit

Indexed keywords

EID: 0031321246     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1030741082     Document Type: Article

References (36)

1. BARRON, A., BIRGÉ, L. and MASSART, P. (1997). Risk bounds for model selection via penalization. Probab. Theory Related Fields. To appear.
2. BARRON, A. and COVER, T. (1991). Minimum complexity density estimation. IEEE Trans. Inform. Theory 37 1034-1054.
3. BERAN, R. J. (1977). Minimum Hellinger distance estimates for parametric models. Ann. Statist. 5 445-463.
4. BIRGÉ, L. (1983). Approximation dans les espaces métriques et théorie de l'estimation. Z. Wahrsch. Verw. Gebiete 65 181-237.
5. CHAUDHURI, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. 19 760-777.
6. CHEN, H. (1991). Estimation of a projection-pursuit regression model. Ann. Statist. 19 1142-1157.
7. DEVROYE, L. P. (1987). A Course in Density Estimation. Birkhäuser, Boston.
8. 1-View. Wiley, New York.
9. DONOHO, D. L., JOHNSTONE I. M., KERKYACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: asymptopia (with discussion)? J. Roy. Statist. Soc. Ser. B 57 301-369.
10. DONOHO, D. L. and LIU, R. C. (1988a). The *automatic* robustness of minimum distance functionals. Ann. Statist. 16 552-586.
11. DONOHO, D. L. and LIU, R. C. (1988b). Pathologies of some minimum distance estimators. Ann. Statist. 16 587-608.
12. FISHER, R. A. (1922). On the mathematical foundations of theoretical statistics. Philos. Trans. Roy. Soc. A 222 309-368.
13. FISHER, R. A. (1925). Theory of statistical estimation. Proc. Cambridge Philos. Soc. 22 700-725.
14. FRIEDMAN, J. H. and STUETZLE, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817-823.
15. GRENANDER, U. (1981). Abstract Inference. Wiley, New York.
16. HOEFFDING, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-31.
17. HUBER, P. J. (1985). Projection pursuit. Ann. Statist. 13 435-475.
18. KOLMOGOROV, A. N. and TIKHOMIHOV, V. M. (1959). ε-entropy and ε-capacity of sets in function spaces. Uspekhi Mat. Nauk. 14 3-86. (In Russian.) [Published in English in (1961) Amer. Math. Soc. Transl. (2) 17 277-364.]
19. LE CAM, L. M. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38-53.
20. LE CAM, L. M. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
21. LE CAM, L. M. and YANG, G. L. (1990). Asymptotics in Statistics: Some Basic Concepts. Springer, New York.
22. MILLAR, P. W. (1982). Robust estimation via minimum distance methods. Z. Warsch. Verw. Gebiete 55 73-89.
23. NICOLERIS, T. (1994). Selected topics in estimation. Ph.D. dissertation, Univ. Montréal.
24. ROUSSAS, G. G. and YATRACOS, Y. G. (1996). Minimum distance regression-type estimates with rates under weak dependence. Ann. Inst. Statist. Math. 48 267-281.
25. ROUSSAS, G. G. and YATRACOS, Y. G. (1997). Minimum distance estimates with rates under φ-mixing. In Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics (D. Pollard, E. Torgersen and G. L. Yang, eds.) 337-345. Springer, New York.
26. STONE, C. J. (1982). Optimal global rates of convergence in nonparametric regression. Ann. Statist. 10 1040-1053.
27. STONE, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689-705.
28. STONE, C. J. (1994). The use of polynomial splines and their tensor product in multivariate function estimation. Ann. Statist. 22 118-184.
29. TRUONG, Y. K. (1989). Asymptotic properties of kernel estimators based on local medians Ann. Statist. 17 606-617.
30. TRUONG, Y. K. and STONE, C. J. (1994). Semi-parametric time series regression. J. Time Ser. Anal. 15 405-428.
31. WOLFOWITZ, J. (1957). The minimum distance method. Ann. Math. Statist. 28 75-88.
32. YATRACOS, Y. G. (1985). Rates of convergence of minimum distance estimators and Kolmogorov's entropy. Ann Statist. 13 768-774.
33. YATRACOS, Y. G. (1988). A lower bound on the error in nonparametric regression type problems. Ann. Statist. 16 1180-1187.
34. YATRACOS, Y. G. (1989a). A regression type problem. Ann. Statist. 17 1597-1607.
35. YATRACOS, Y. G. (1989b). On the estimation of the derivatives of a function via the derivatives of an estimate. J. Multivariate Anal. 28 172-175.
36. d. J. Multivariate Anal. 40 213-221.