# zbMATH — the first resource for mathematics

Dimension reduction for regression estimation with nearest neighbor method. (English) Zbl 1329.62254
Summary: In regression with a high-dimensional predictor vector, dimension reduction methods aim at replacing the predictor by a lower dimensional version without loss of information on the regression. In this context, the so-called central mean subspace is the key of dimension reduction. The last two decades have seen the emergence of many methods to estimate the central mean subspace. In this paper, we go one step further, and we study the performances of a $$k$$-nearest neighbor type estimate of the regression function, based on an estimator of the central mean subspace. In our setting, the predictor lies in $$\mathbb R^{p}$$ with fixed $$p$$, i.e. it does not depend on the sample size. The estimate is first proved to be consistent. Improvement due to the dimension reduction step is then observed in term of its rate of convergence. All the results are distributions-free. As an application, we give an explicit rate of convergence using the SIR method. The method is illustrated by a simulation study.

##### MSC:
 62H12 Estimation in multivariate analysis 62G08 Nonparametric regression and quantile regression
Full Text:
##### References:
 [1] Cook, R.D. (1994). On the Interpretation of Regression Plot., Journal of the American Statistical Association 89 177-189. · Zbl 0791.62066 [2] Cook, R.D. (1996). Graphics for Regression with a Binary Response., Journal of the American Statistical Association 91 , 983-992. · Zbl 0882.62060 [3] Cook, R.D. (1998)., Regression Graphics . Wiley, New-York NY. [4] Cook, R.D. and Li, B. (2002). Dimension Reduction for Conditional Mean in Regression, The Annals of Statistics 30 , 455-474. · Zbl 1012.62035 [5] Cook, R.D. and Ni, L. (2005). Sufficient Dimension Reduction via Inverse Regression: A Minimum Discrepancy Approach., Journal of the American Statistical Association 100 , 410-428. · Zbl 1117.62312 [6] Cook, R.D. and Weisberg, S. (1991). Discussion of “Sliced Inverse Regression for Dimension Reduction”., Journal of the American Statistical Association 86 , 316-342. · Zbl 1353.62037 [7] Cook, R.D. and Weisberg, S. (1999). Graphs in Statistical Analysis: Is the Medium the Message?, The American Statistician 53 , 29-37. [8] Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002)., A Distribution-Free Theory of Nonparametric Regression . Springer-Verlag, New-York NY. · Zbl 1021.62024 [9] Hall, P. and Li, K.C. (1993). On Almost Linearity of Low-Dimensional Projections from High-Dimensional Data., The Annals of Statistics 21 , 867-889. · Zbl 0782.62065 [10] Härdle, W. and Stoker, T.M. (1989). Investigating Smooth Multiple Regression by the Method of Average Derivative., Journal of the American Statistical Association 84 , 986-995. · Zbl 0703.62052 [11] Ibragimov, I.A. and Khasminskii, R.Z. (1981)., Statistical Estimation: Asymptotic Theory . Springer-Verlag, New-York NY. [12] Kato, T. (1966)., Perturbation Theory for Linear Operators . Springer-Verlag, New-York NY. · Zbl 0148.12601 [13] Li, K.C. (1991). Sliced Inverse Regression for Dimension Reduction (with Discussion)., Journal of the American Statistical Association 86 , 316-342. · Zbl 0742.62044 [14] Li, K.C. (1992). On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein’s Lemma., Journal of the American Statistical Association 87 , 1025-1039. · Zbl 0765.62003 [15] Saracco, J. (2005). Asymptotics for Pooled Marginal Slicing Estimator Based on, SIR \alpha . Journal of Multivariate Analysis 96 , 117-135. · Zbl 1080.62008 [16] Xia, Y., Tong, H., Li, W.K. and Zhu, L.-X. (2002). An Adaptative Estimation of Dimension Reduction Space., Journal of the Royal Statistical Society, Ser. B 64 , 1-28. · Zbl 1091.62028 [17] Ye, Z. and Weiss, R.E. (2003). Using the Bootstrap to Select One of a New Class of Dimension Reduction Methods., Journal of the American Statistical Association 98 , 968-979. · Zbl 1045.62034 [18] Zhu, L., Miao, B. and Peng, H. (2006). On Sliced Inverse Regression with High-Dimensional Covariates., Journal of the American Statistical Association 101 , 630-643. · Zbl 1119.62331 [19] Zhu, Y. and Zeng P. (2006). Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression., Journal of the American Statistical Association 101 , 1638-1651. · Zbl 1171.62325
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.