# zbMATH — the first resource for mathematics

Dimension reduction for nonelliptically distributed predictors. (English) Zbl 1160.62050
Summary: Sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor, or, when such conditions are not satisfied, rely on marginal transformations or reweighting to fulfill them approximately. For example, a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distributions. We reformulate the commonly used dimension reduction methods, via the notion of “central solution space” to circumvent the requirements of such strong assumptions, while at the same time preserve the desirable properties of the classical methods, such as $$\sqrt n$$-consistency and asymptotic normality. Imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the “curse of dimensionality”, but the developments of this paper show that this need not be the case. The new methods will be compared with existing methods by simulation and applied to a data set.

##### MSC:
 62H12 Estimation in multivariate analysis 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62G09 Nonparametric statistical resampling methods 62E20 Asymptotic distribution theory in statistics 65C60 Computational problems in statistics (MSC2010)
Full Text:
##### References:
 [1] Allen, D. M. (1974). The relationship between variable selection and data augmentation and a method for prediction. Technometrics 16 125-127. JSTOR: · Zbl 0286.62044 [2] Bellman, R. (1961). Adaptive Control Processes : A Guided Tour . Princeton Univ. Press. · Zbl 0103.12901 [3] Bickel, P., Klaassen, C. A. J., Ritov, Y. and Wellner, J. (1993). Efficient and Adaptive Inference in Semi-Parametric Models . Johns Hopkins Univ. Press, Baltimore. · Zbl 0786.62001 [4] Bura, E. and Cook, R. D. (2001). Estimating the structural dimension of regressions via parametric inverse regression. J. Roy. Statist. Soc. Ser. B 63 393-410. JSTOR: · Zbl 0979.62041 [5] Chiaromonte, F. and Cook, R. D. (2001). Sufficient dimension reduction and graphics in regression. Ann. Inst. Statist. Math. 54 768-795. · Zbl 1047.62066 [6] Cook, R. D. (1994). Using dimension-reduction subspaces to identify important inputs in models of physical systems. In Amer. Statist. Assoc. Proceedings of the Section on Physical and Engineering Sciences . Amer. Statist. Assoc., Washington, DC. [7] Cook, R. D. (1996). Graphics for regressions with a binary response. J. Amer. Statist. Assoc. 91 983-992. JSTOR: · Zbl 0882.62060 [8] Cook, R. D. (1998). Regression Graphics : Ideas for Studying Regressions Through Graphics . Wiley, New York. · Zbl 0903.62001 [9] Cook, R. D. (2007). Fisher lecture: Dimension reduction for regression (with discussion). Statist. Sci. 22 1-26. · Zbl 1246.62148 [10] Cook, R. D. and Nachtsheim, C. J. (1994). Reweighting to achieve elliptically contoured covariates in regression. J. Amer. Statist. Assoc. 89 592-599. · Zbl 0799.62078 [11] Cook, R. D. and Ni, L. (2005). Sufficient dimension reduction via inverse regression: A minimum discrepancy approach. J. Amer. Statist. Assoc. 100 410-428. · Zbl 1117.62312 [12] Cook, R. D. and Ni, L. (2006). Using intra-slice covariances for improved estimation of central subspace in regression. Biometrika 93 65-74. · Zbl 1152.62019 [13] Cook, R. D. and Weisberg, S. (1991). Discussion of “Sliced inverse regression for dimension reduction,” by K.-C. Li. J. Amer. Statist. Assoc. 86 328-332. JSTOR: · Zbl 0742.62044 [14] Dawid, A. P. (1979). Conditional independence in statistical theory (with discussion). J. Roy. Statist. Soc. Ser. B 41 1-31. JSTOR: · Zbl 0408.62004 [15] Eaton, M. L. (1986). A characterization of spherical distributions. J. Multivariate Anal. 34 439-446. · Zbl 0596.62057 [16] Fernholz, L. T. (1983). Von Mises Calculus for Statistical Functionals . Springer, New York. · Zbl 0525.62031 [17] Ferre, L. and Yao, A. F. (2005). Smooth function inverse regression. Statist. Sinica 15 665-683. · Zbl 1086.62054 [18] Fung, K. F., He, X., Liu, L. and Shi, P. (2002). Dimension reduction based on canonical correlation. Statist. Sinica 12 1093-1113. · Zbl 1004.62058 [19] Hall, W. J. and Mathiason, D. J. (1990). On large-sample estimation and testing in parametric models. Internat. Statist. Rev. 58 77-97. · Zbl 0715.62058 [20] Li, B. and Wang, S. (2007). On directional regression for dimension reduction. J. Amer. Statist. Assoc. 102 997-1008. · Zbl 05564427 [21] Li, K. C. (1991). Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc. 86 316-342. JSTOR: · Zbl 0742.62044 [22] 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 [23] Li, K. C. and Duan, N. (1989). Regression analysis under link violation. Ann. Statist. 17 1009-1052. · Zbl 0753.62041 [24] McCullagh, P. (1987). Tensor Methods in Statistics . Chapman and Hall, London. · Zbl 0732.62003 [25] Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization. Comput. J. 7 308-313. · Zbl 0229.65053 [26] Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions. J. Roy. Statist. Soc. Ser. B 36 111-147. JSTOR: · Zbl 0308.62063 [27] van der Vaart, A. W. (1998). Asymptotic Statistics . Cambridge Univ. Press. · Zbl 0910.62001 [28] von Mises, R. (1947). On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statist. 18 309-348. · Zbl 0037.08401 [29] Xia, Y., Tong, H., Li, W. K. and Zhu, L. X. (2002). An adaptive estimation of optimal regression subspace. J. Roy. Statist. Soc. Ser. B 64 363-410. JSTOR: · Zbl 1091.62028 [30] Yin, X., Li, B. and Cook, R. D. (2008). Successive direction extraction for estimating the central subspace in a multiple-index regression. J. Multivariate Anal. 99 1733-1757. · Zbl 1144.62030 [31] Zhu, L.-X. and Fang, K.-T. (1996). Asymptotics for kernel estimate of sliced inverse regression. Ann. Statist. 3 1053-1068. · Zbl 0864.62027
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.