An adaptive estimation of dimension reduction space

Journal of the Royal Statistical Society. Series B: Statistical Methodology

Volumn 64, Issue 3, 2002, Pages 363-410

  Xia, Yingcun ^a   Tong, Howell ^a   Li, W K ^a   Zhu, Li Xing ^a  

^a LONDON SCHOOL OF ECONOMICS   (United Kingdom)

Author keywords

Average derivative estimation; Dimension reduction; Generalized linear models; Local linear smoother; Multiple time series; Non linear time series analysis; Nonparametric regression; Principal Hessian direction; Projection pursuit; Semiparametrics

Indexed keywords

EID: 0036428498     PISSN: 13697412     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9868.03411     Document Type: Article

References (37)

1. Auestad, B. and Tjøstheim, D. (1990) Identification of nonlinear time series: first order characterization and order determination. Biometrika, 77, 669-688.
2. Breiman, L., Friedman, J.H., Olshen, R.A. and Stone, C.J. (1984) Classification and Regression Trees. Belmont: Wadsworth.
3. Cai, Z., Fan, J. and Yao, Q. (2000) Functional-coefficient regression models for nonlinear time series. J. Am. Statist. Ass., 95, 941-956.
4. Carroll, R.J., Fan, J., Gijbels, I. and Wand, M.P. (1997) Generalized partially linear single-index models. J. Am. Statist. Ass., 92, 477-489.
5. Carroll, R.J. and Li, K.C. (1995) Binary regressors in dimension reduction models: a new look at treatment comparisons. Statist. Sin., 5, 667-688.
6. Chaudhuri, P., Huang, M.C., Loh, W.Y. and Yao, R. (1994) Piecewise-polynomial regression trees. Statist. Sin., 4, 143-167.
7. Chen, C.-H. and Li, K.C. (1989) Can SIR be as popular as multiple linear regression? Statist. Sin., 8, 289-316.
8. Chen, H. (1991) Estimation of a projection-pursuit type regression model. Ann. Statist., 19, 142-157.
9. Cheng, B. and Tong, H. (1992) On consistent nonparametric order determination and chaos (with discussion). J. R. Statist. Soc. B, 54, 427-449.
10. Cook, R.D. (1994) On the interpretation of regression plots. J. Am. Statist. Ass., 89, 177-189.
11. _ (1998) Principal Hessian directions revisited (with discussions). J. Am. Statist. Ass., 93, 85-100.
12. Fan, J. and Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. London: Chapman and Hall.
13. Fan, J. and Zhang, W.Y. (1999) Statistical estimation in varying coefficient models. Ann. Statist., 27, 1491-1518.
14. Friedman, J.H. and Stuetzle, W. (1981) Projection pursuit regression. J. Am. Statist. Ass., 76, 817-823.
15. Hall, P. (1989) On projection pursuit regression. Ann. Statist., 17, 573-588.
16. Hannan, E.J. (1969) The estimation of mixed moving average autoregressive system. Biometrika, 56, 579-593.
17. Härdle, W., Hall, P. and Ichimura, H. (1993) Optimal smoothing in single-index models. Ann. Statist., 21, 157-178.
18. Härdle, W. and Stoker, T.M. (1989) Investigating smooth multiple regression by method of average derivatives. J. Am. Statist. Ass., 84, 986-995.
19. Hastie, T.J. and Tibshirani, R. (1986) Generalized additive models (with discussion). Statist. Sci., 1, 297-318.
20. Hotelling, H. (1935) The most predictable criterion. J. Educ. Psychol., 26, 139-142.
21. Huber, P.J. (1985) Projection pursuit (with discussion). Ann. Statist., 13, 435-525.
22. Ichimura, H. and Lee, L. (1991) Semiparametric least squares estimation of multiple index models: single equation estimation. In Nonparametric and Semiparametric Methods in Econometrics and Statistics (eds W. Barnett, J. Powell and G. Tauchen). Cambridge: Cambridge University Press.
23. Li, K.C. (1991) Sliced inverse regression for dimension reduction (with discussion). J. Am. Statist. Ass., 86, 316-342.
24. _ (1992) On principal Hessian directions for data visualisation and dimension reduction: another application of Stein's Lemma. Ann. Statist., 87, 1025-1039.
25. Li, K.C., Lue, H.H. and Chen, C.H. (2000) Interactive tree-structured regression via principal Hessian directions. J. Am. Statist. Ass., 95, 547-560.
26. Rao, C.R. (1973) Linear Statistical Inference and Its Applications. New York: Wiley.
27. Schott, J.R. (1994) Determining the dimensionality in sliced inverse regression. J. Am. Statist. Ass., 89, 141-148.
28. Severini, T.A. and Wong, W.H. (1992) Profile likelihood and conditionally parametric models. Ann. Statist., 20, 1768-1802.
29. Smith, R.L., Davis, J.M. and Speckman, P. (1999) Assessing the human health risk of atmospheric particles. In Environmental Statistics: Analysing Data for Environmental Policy. New York: Wiley.
30. Stone, M. (1974) Cross-validatory choice and assessment of statistical predictions (with discussion). J. R. Statist. Soc. B, 36, 111-147.
31. Tong, H. (1990) Nonlinear Time Series Analysis: a Dynamical System Approach. Oxford: Oxford University Press.
32. Xia, Y. and An, H.-Z. (1999) Projection pursuit autoregression in time series. J. Time Ser. Anal., 20, 693-714.
33. Xia, Y. and Li, W.K. (1999) On single-index coefficient regression models. J. Am. Statist. Ass., 94, 1275-1285.
34. Xia, Y., Tong, H. and Li, W.K. (1999) On extended partially linear single-index models. Biometrika, 86, 831-842.
35. Yang, L. and Tschernig, R. (1999) Multivariate bandwidth selection for local linear regression. J. R. Statist. Soc. B, 61, 793-815.
36. Yao, Q. and Tong, H., (1994) On subset selection in nonparametric stochastic regression. Statist. Sin., 4, 51-70.
37. Zhu, L.X. and Fang, K.-T. (1996) Asymptotics for kernel estimate of sliced inverse regression. Ann. Statist., 24, 1053-1068.