Hsing, Tailen Nearest neighbor inverse regression. (English) Zbl 0951.62034 Ann. Stat. 27, No. 2, 697-731 (1999). Summary: Sliced inverse regression (SIR), formally introduced by K.-C. Li [J. Am. Stat. Assoc. 86, No. 414, 316-342 (1991; Zbl 0742.62044)], 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. Cited in 24 Documents MSC: 62G08 Nonparametric regression and quantile regression 60F05 Central limit and other weak theorems 62G20 Asymptotic properties of nonparametric inference Keywords:central limit theorem; sliced inverse regression; dimension reduction Citations:Zbl 0742.62044 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Aldous, D. and Steele, M. (1992). Asymptotics for Euclidean minimal spanning trees on random graphs. Probab. Theory Related Fields 92 247-258. · Zbl 0767.60005 · doi:10.1007/BF01194923 [2] Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033-1046. · Zbl 0784.60015 · doi:10.1214/aoap/1177005271 [3] 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. [4] 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. · Zbl 0502.62045 · doi:10.1214/aop/1176993668 [5] Breiman, L., Friedman, J. H., Olshen, R. and Stone, C. (1984). Classification of Regression Trees. Wadsworth, Belmont, CA. · Zbl 0541.62042 [6] Chen, H. (1991). Estimation of a projection-pursuit type regression model. Ann. Statist. 19 142- 157. · Zbl 0736.62055 · doi:10.1214/aos/1176347974 [7] Cook, R. D. (1995). Graphics for studying net effects of regression predictors. Statist. Sinica 5 689-708. · Zbl 0824.62063 [8] Cook, R. D. (1998). Principal Hessian directions revisited. J. Amer. Statist. Assoc. 93 84-100. JSTOR: · Zbl 0922.62057 · doi:10.2307/2669605 [9] Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817-823. JSTOR: · doi:10.2307/2287576 [10] Hall, P. (1989). On projection pursuit regression. Ann. Statist. 17 573-588. · Zbl 0698.62041 · doi:10.1214/aos/1176347126 [11] Hall, P. and Li, K. C. (1993). On almost linearity of low-dimensional projections from highdimensional data. Ann. Statist. 21 867-889. · Zbl 0782.62065 · doi:10.1214/aos/1176349155 [12] Härdle, W. and Stoker, T. M. (1989). Investigation smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84 986-995. · Zbl 0703.62052 [13] Hastie, T. and Tibshirani, R. (1986). Generalized additive models. Statist. Sci. 1 297-318. · Zbl 0955.62603 · doi:10.1214/ss/1177013604 [14] Huber, P. (1985). Projection pursuit (with discussion). Ann. Statist. 13 435-526. · Zbl 0595.62059 · doi:10.1214/aos/1176349519 [15] Li, K. C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316-342. JSTOR: · Zbl 0742.62044 · doi:10.2307/2290563 [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. JSTOR: · Zbl 0765.62003 · doi:10.2307/2290640 [17] 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. [18] Samorov, A. M. (1993). Exploring regression structure using nonparametric functional estimation. J. Amer. Statist. Assoc. 88 836-847. JSTOR: · Zbl 0790.62035 · doi:10.2307/2290772 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.