Structure adaptive approach for dimension reduction

Annals of Statistics

Volumn 29, Issue 6, 2001, Pages 1537-1566

  Hristache, Marian ^a   Juditsky, Anatoli ^b   Polzehl, Jörg ^c   Spokoiny, Vladimir ^c  

^a ENSAI Ecole Nationale de la Statistique et de lAnalyse de lInformation   (France)

^b Weierstrass Institute ^*  (France)

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

Author keywords

Average derivative estimation; Dimension reduction; Index space; Multi index model; Structural adaptation

Indexed keywords

EID: 0035528162     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1015345954     Document Type: Article

References (18)

1. BRILLINGER, D. R. (1983). A generalized linear model with "Gaussian" regressor variables. In A Festschrift for Eric L. Lehmann (P. J. Bickel, K. A. Doksum and J. A. Hodges, eds.) 97-114. Wadsworth, Belmont, CA.
2. CARROLL, R. J., FAN, J., GIJBELS, I. and WAND, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477-489.
3. COOK, D. (1998). Principal Hessian directions revisited. J. Amer. Statist. Assoc. 93 84-93.
4. DOKSUM, K. and SAMAROV, A. (1995). Nonparametric estimation of global functionals and a measure of explanatory power of covariates in regression. Ann. Statist. 23 1443-1473.
5. DONOHO, D. L. and NUSSBAUM, M. (1990). Minimax quadratic estimation of a quadratic functional. J. Complexity 6 290-323.
6. FAN, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273-1294.
7. FAN, J. and GIJBELS, X. (1996). Local Polynomial Modelling and Its Applications. Chapman & Hall, London.
8. HORN, R. A. and JOHNSON, C. R. (1985). Matrix Analysis. Cambridge Univ. Press.
9. HARVILLE, D. A. (1997). Matrix Analysis from a Statistitian's Perspective. Springer, New York.
10. HRISTACHE, M., JUDITSKY, A. and SPOKOINY, V. (2001). Direct estimation of the index coefficients in a single-index model. Ann. Statist. 29 1-32.
11. HUANG, L.-S. and FAN, J. (1998). Nonparametric estimation of quadratic regression functionals. Benoulli 5 927-949.
12. IBRAGIMOV, I., NEMIROVSKII, A. and KHASMINSKI, R. (1986). Some problems on noparametric estimation in Gaussian white noise. Theory Probab. Appl. 31 391-406.
13. KING, M. S. (1997). Local likelihood and local partial likelihood in hazard regression. Ph.D. dissertation, Univ. North Carolina.
14. LI, K-C. and DUAN, N. (1989). Regression analysis under link violation. Ann. Statist. 17 1009-1052.
15. LI, K-C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316-342.
16. LI, K-C. (1992). On principal Hessian directions for data visualization and dimension reduction: another application of Stein's lemma. J. Amer. Statist. Assoc. 87 1025-1039.
17. LI, K.-C. (2000). High dimensional data analysis via the SIR/PHD approach. Available at http://www.stat.ucla.edu/ kcli/sir-PHD.ps.gz.
18. SAMAROV, A. (1993). Exploring regression structure using nonparametric functional estimation. J. Amer. Statist. Assoc. 88 836-847.