The Annals of Statistics

Nearest neighbor inverse regression

Tailen Hsing

Full-text: Open access

Abstract

Sliced inverse regression (SIR), formally introduced by Li, is a very general procedure for performing dimension reduction in nonparametric regression. This paper considers a version of SIR in which the “slices” are determined by nearest neighbors and the response variable takes value possibly in a multidimensional space. It is shown, under general conditions, that the “effective dimension reduction space” can be estimated with rate $n^{-1/2}$ where $n$ is the sample size.

Article information

Source
Ann. Statist. Volume 27, Number 2 (1999), 697-731.

Dates
First available in Project Euclid: 5 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1018031213

Digital Object Identifier
doi:10.1214/aos/1018031213

Mathematical Reviews number (MathSciNet)
MR1714711

Zentralblatt MATH identifier
0951.62034

Subjects
Primary: 62G05: Estimation
Secondary: 62F05: Asymptotic properties of tests

Keywords
Central limit theorem dimension reduction nonparametric regression sliced inverse regression.

Citation

Hsing, Tailen. Nearest neighbor inverse regression. Ann. Statist. 27 (1999), no. 2, 697--731. doi:10.1214/aos/1018031213. http://projecteuclid.org/euclid.aos/1018031213.


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References

  • Aldous, D. and Steele, M. (1992). Asymptotics for Euclidean minimal spanning trees on random graphs. Probab. Theory Related Fields 92 247-258.
  • Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033-1046.
  • Bai, Z. D., Miao, B. Q. and Radhakrishna, R. (1991). Estimation of directions of arrival of signals: asymptotic results. In Advances in Spectral Analysis and Array Processing (S. Haykin, ed.) 2 327-347. Prentice-Hall, Englewood Cliffs, NJ.
  • Bickel, P. J. and Breiman, L. (1983). Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11 185-214.
  • Breiman, L., Friedman, J. H., Olshen, R. and Stone, C. (1984). Classification of Regression Trees. Wadsworth, Belmont, CA.
  • Chen, H. (1991). Estimation of a projection-pursuit type regression model. Ann. Statist. 19 142- 157.
  • Cook, R. D. (1995). Graphics for studying net effects of regression predictors. Statist. Sinica 5 689-708.
  • Cook, R. D. (1998). Principal Hessian directions revisited. J. Amer. Statist. Assoc. 93 84-100.
  • Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817-823.
  • Hall, P. (1989). On projection pursuit regression. Ann. Statist. 17 573-588.
  • Hall, P. and Li, K. C. (1993). On almost linearity of low-dimensional projections from highdimensional data. Ann. Statist. 21 867-889.
  • H¨ardle, W. and Stoker, T. M. (1989). Investigation smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84 986-995.
  • Hastie, T. and Tibshirani, R. (1986). Generalized additive models. Statist. Sci. 1 297-318.
  • Huber, P. (1985). Projection pursuit (with discussion). Ann. Statist. 13 435-526.
  • Li, K. C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316-342.
  • 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.
  • Li, K. C., Aragon, Y. and Thomas-Agnan, C. (1994). Analysis of multivariate outcome data: SIR and a non-linear theory of Hotelling's most predictable variates. Preprint.
  • Samorov, A. M. (1993). Exploring regression structure using nonparametric functional estimation. J. Amer. Statist. Assoc. 88 836-847.