Cook, R. Dennis; Li, Bing Dimension reduction for conditional mean in regression. (English) Zbl 1012.62035 Ann. Stat. 30, No. 2, 455-474 (2002). Summary: In many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors, and less concerned with the other aspects of the conditional distribution. We develop dimension reduction methods that incorporate this consideration. We introduce the notion of the Central Mean Subspace (CMS), a natural inferential object for dimension reduction when the mean function is of interest. We study properties of the CMS, and develop methods to estimate it. These methods include a new class of estimators which requires fewer conditions than pHd, and which displays a clear advantage when one of the conditions for pHd is violated. CMS also reveals a transparent distinction among the existing methods for dimension reduction: OLS, pHd, SIR and SAVE. We apply the new methods to a data set involving recumbent cows. Cited in 175 Documents MSC: 62G08 Nonparametric regression and quantile regression 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62A09 Graphical methods in statistics × Cite Format Result Cite Review PDF Full Text: DOI References: [1] 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 · doi:10.1111/1467-9868.00292 [2] CHIAROMONTE, F., COOK, R. D. and LI, B. (2002). Sufficient dimension reduction in regressions with categorical predictors. Ann. Statist. 30 475-497. · Zbl 1012.62036 · doi:10.1214/aos/1021379862 [3] CLARK, R. G., HENDERSON, H. V., HOGGARD, G. K. ELLISON, R. S. and YOUNG, B. J. (1987). The ability of biochemical and haematological tests to predict recovery in periparturient recumbent cows. New Zealand Veterinary Journal 35 126-133. [4] COOK, R. D. (1992). Regression plotting based on quadratic predictors. In L1-Statistical Analysis and Related Methods (Y. Dodge, ed.) 115-127. North-Holland, Amsterdam. [5] COOK, R. D.(1994a). On the interpretation of regression plots. J. Amer. Statist. Assoc. 89 177-189. · Zbl 0791.62066 [6] COOK, R. D. (1994b). Using dimension-reduction subspaces to identify important inputs in models of physical systems. In Proceedings of the Section on Physical and Engineering Sciences 18-25. Amer. Statistic. Assoc., Alexandria, VA. [7] COOK, R. D. (1996). Graphics for regressions with a binary response. J. Amer. Statist. Assoc. 91 983-992. JSTOR: · Zbl 0882.62060 · doi:10.2307/2291717 [8] COOK, R. D.(1998a). Regression Graphics. Wiley, New York. · Zbl 0903.62001 [9] COOK, R. D. (1998b). Principal Hessian directions revisited. J. Amer. Statist. Assoc. 93 84-100. · Zbl 0922.62057 [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 · doi:10.2307/2290862 [11] COOK, R. D. and WEISBERG, S. (1983). Diagnostics for heteroscedasticity in regression. Biometrika 70 1-10. JSTOR: · Zbl 0502.62063 · doi:10.1093/biomet/70.1.1 [12] COOK, R. D. and WEISBERG, S. (1991). Discussion of ”Sliced inverse regression for dimension reduction.” J. Amer. Statist. Assoc. 86 328-332. · Zbl 0949.47013 [13] COOK, R. D. and WEISBERG, S. (1999). Graphics in statistical analysis: Is the medium the message? The American Statistician 53 29-37. [14] DIACONIS, P. and FREEDMAN, D. (1984). Asymptotics of graphical projection pursuit. Ann. Statist. 12 793-815. · Zbl 0559.62002 · doi:10.1214/aos/1176346703 [15] EATON, M. L. and TYLER, D. (1994). The asymptotic distribution of singular values with application to canonical correlations and correspondence analysis. J. Multivariate Anal. 50 238-264. · Zbl 0805.62020 · doi:10.1006/jmva.1994.1041 [16] HALL, P. and LI, K.-C. (1993). On almost linearity of low dimensional projections from high dimensional data. Ann. Statist. 21 867-889. · Zbl 0782.62065 · doi:10.1214/aos/1176349155 [17] 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 [18] 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 [19] LI, K.-C. and DUAN, N. (1989). Regression analysis under link violation. Ann. Statist. 17 1009- 1952. · Zbl 0753.62041 · doi:10.1214/aos/1176347254 [20] UNIVERSITY PARK, PENNSYLVANIA 16802 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.