The Annals of Statistics

Asymptotics for kernel estimate of sliced inverse regression

Kai-Tai Fang and Li-Xing Zhu

Full-text: Open access

Abstract

To explore nonlinear structures hidden in high-dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression, Li and Duan proposed the sliced inverse regression (SIR) method which is simple to use. In this paper, the asymptotic properties of the kernel estimate of sliced inverse regression are investigated. It turns out that regardless of the kernel function, the asymptotic distribution remains the same for a wide range of smoothing parameters.

Article information

Source
Ann. Statist., Volume 24, Number 3 (1996), 1053-1068.

Dates
First available in Project Euclid: 20 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032526955

Digital Object Identifier
doi:10.1214/aos/1032526955

Mathematical Reviews number (MathSciNet)
MR1401836

Zentralblatt MATH identifier
0864.62027

Subjects
Primary: 60F05: Central limit and other weak theorems 62G05: Estimation 62J02: General nonlinear regression

Keywords
Data structure dimension reduction kernel estimation nonparametric regression sliced inverse regression

Citation

Zhu, Li-Xing; Fang, Kai-Tai. Asymptotics for kernel estimate of sliced inverse regression. Ann. Statist. 24 (1996), no. 3, 1053--1068. doi:10.1214/aos/1032526955. https://projecteuclid.org/euclid.aos/1032526955


Export citation

References

  • DUAN, N. and LI, K. C. 1991. Slicing regression: a link free regression method. Ann. Statist. 19 505 530. Z.
  • FRIEDMAN, J. H. and STUETZLE, W. 1981. Projection pursuit regression. J. Amer. Statist. Assoc. 76 817 823. Z.
  • FRIEDMAN, J. H. and TUKEY, J. W. 1974. A projection pursuit algorithm for exploratory data analysis. IEEE Trans. Comput. C-23 881 889. Z.
  • HALL, P. 1989. On projection pursuit regression. Ann. Statist. 17 583 588. Z.
  • HARDLE, W. and STOKER, T. M. 1989. Investigating smooth multiple regression by the method ¨ of average derivatives. J. Amer. Statist. Assoc. 84 986 995. Z.
  • HASTIE, T. and TIBSHIRANI, R. 1986. Generalized additive models. Statist. Sci. 1 297 318. Z.
  • HSING, T. and CARROLL, R. J. 1992. An asy mptotic theory for sliced inverse regression. Ann. Statist. 20 1040 1061. Z. Z.
  • HUBER, P. 1985. Projection pursuit with discussion. Ann. Statist. 13 435 475. Z.
  • KATO, T. 1983. Perturbation Theory for Linear Operation, 2nd ed. Springer, New York. Z.
  • LI, K. C. 1989. Data visualization with SIR: a transformation based projection pursuit method. UCLA Statist. Ser. 24. Z. Z.
  • LI, K. C. 1991. Sliced inverse regression for dimension reduction with discussion. J. Amer. Statist. Assoc. 86 316 342. Z.
  • NOLAN, D. and POLLARD, D. 1987. U-processes: rates of convergence. Ann. Statist. 15 780 799. Z.
  • POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z.
  • POWELL, J. L., STOCK, J. H. and STOKER, T. M. 1989. Semiparametric estimation of index coefficients. Econometrica 57 1403 1430. Z.
  • RAO, B. L. S. P. 1983. Nonparametric Functional Estimation. Academic Press, Orlando, FL. Z.
  • STONE, C. J. 1984. An asy mptotically optimal window selection rule for kernel density estimate. Ann. Statist. 12 1285 1297. Z.
  • SUN, S. G. 1988. Analy tic expressions for the derivatives of the eigenvalues and eigenvectors of Z. Z. a matrix. Adv. in Math. Beijing 17 391 397 in Chinese. Z.
  • Ty LER, D. 1981. Asy mptotic inference for eigenvectors. Ann. Statist. 9 725 736. Z.
  • ZHU, L. X. 1993. Convergence rates of the empirical processes indexed by the classes of Z. functions with applications. J. Sy stems Sci. Math. Sci. 13 33 41 in Chinese. Z.
  • ZHU, L. X. and FANG, K. T. 1992. Projection pursuit approximation for nonparametric regresZ sion. Proceedings of the Order Statistic and Nonparametric: Theory and Methods P.. K. Sen and I. A. Salama, eds. 455 469. North-Holland, Amsterdam.
  • BEIJING, 100080 KOWLOON CHINA HONG KONG