In this article, we introduce a flexible model-free approach to sufficient dimension reduction analysis using the expectation of conditional difference measure. Without any strict conditions, such as linearity condition or constant covariance condition, the method estimates the central subspace exhaustively and efficiently under linear or nonlinear relationships between response and predictors. The method is especially meaningful when the response is categorical. We also studied the n½-consistency and asymptotic normality of the estimate. The efficacy of our method is demonstrated through both simulations and a real data analysis.

• Bellman, R. (1961). Adaptive control processes. Princeton University Press. 
• Byrd, R. H., Gilbert, J. C., & Nocedal, J. (2000). A trust region method based on interior point techniques for nonlinear programming. Mathematical Programming, 89(1), 149–185. https://doi.org/10.1007/PL00011391 
• Byrd, R. H., Mary, E. H., & Nocedal, J. (1999). An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization, 9(4), 877–900. https://doi.org/10.1137/S1052623497325107 
• Cook, R. D. (1994). Using dimension-reduction subspaces to identify important inputs in models of physical systems. Proc. Phys. Eng. Sci. Sect. (pp. 18–25). 
• Cook, R. D. (1996). Graphics for regressions with a binary response. Journal of the American Statistical Association, 91(435), 983–992. https://doi.org/10.1080/01621459.1996.10476968
• Cook, R. D. (1998). Regression graphics: ideas for studying regressions through graphics. Wiley. 
• Cook, R. D., & Forzani, L. (2009). Likelihood-Based sufficient dimension reduction. Journal of the American Statistical Association, 104(485), 197–208. https://doi.org/10.1198/jasa.2009.0106 
• Cook, R. D., & Ni, L. (2005). Sufficient dimension reduction via inverse regression: a minimum discrepancy approach. Journal of the American Statistical Association, 100(470), 410–428. https://doi.org/10.1198/016214504000001501 
• Cook, R. D., & Weisberg, S. (1991). Sliced inverse regression for dimension reduction: comment. Journal of the American Statistical Association, 86(414), 328–332. 
• Cook, R. D., & Zhang, X. (2014). Fused estimators of the central subspace in sufficient dimension reduction. Journal of the American Statistical Association, 109(506), 815–827. https://doi.org/10.1080/01621459.2013.866563 
• Cui, X., Härdle, W., & Zhu, L. (2011). The EFM approach for single-index models. The Annals of Statistics, 12(3), 793–815. 
• Dong, Y., & Li, B. (2010). Dimension reduction for non-elliptically distributed predictors: second-order methods. Biometrika, 97(2), 279–294. https://doi.org/10.1093/biomet/asq016 
• Fung, W., He, X., Liu, L., & Shi, P. (2002). Dimension reduction based on canonical correlation. Statistica Sinica, 12(4), 1093–1113. 
• Härdle, W., & Stoker, T. (1989). Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association, 84(408), 986–995.
• Hristache, M., Juditsky, A., Polzehl, J., & Spokoiny, V. (2001). Structure adaptive approach for dimension reduction. The Annals of Statistics, 29(6), 1537–1811. https://doi.org/10.1214/aos/1015345954 
• Lehmann, E. L. (1999). Elements of large-sample theory. Springer-Verlag
• Li, K.-C. (1991). Sliced inverse regression for dimension reduction. Journal of the American Statistical Association, 86(414), 316–327. https://doi.org/10.1080/01621459.1991.10475035
• 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(420), 1025–1039. https://doi.org/10.1080/01621459.1992.10476258 
• Li, B., & Wang, S. (2007). On directional regression for dimension reduction. Journal of American Statistical Association, 102(479), 997–1008. https://doi.org/10.1198/016214507000000536 
• Li, L., & Yin, X. (2009). Longitudinal data analysis using sufficient dimension reduction method. Computational Statistics and Data Analysis, 53(12), 4106–4115. https://doi.org/10.1016/j.csda.2009.04.018 
• Li, B., Zha, H., & Chiaromonte, F. (2005). Contour regression: a general approach to dimension reduction. The Annals of Statistics, 33(4), 1580–1616. https://doi.org/10.1214/009053605000000192 
• Luo, W., & Li, B. (2016). Combining eigenvalues and variation of eigenvectors for order determination. Biometrika, 103(4), 875–887. https://doi.org/10.1093/biomet/asw051 
• Luo, R., Wang, H., & Tsai, C. L. (2009). Contour projected dimension reduction. The Annals of Statistics, 37(6B), 3743–3778. https://doi.org/10.1214/08-AOS679 
• Ma, Y., & Zhu, L. (2013). Efficient estimation in sufficient dimension reduction. The Annals of Statistics, 41(1), 250–268. https://doi.org/10.1214/12-AOS1072
• Powell, J., Stock, J., & Stoker, T. (1989). Semiparametric estimation of index coefficients. Econometrica: Journal of the Econometric Society, 57(6), 1403–1430. https://doi.org/10.2307/1913713 
• Serfling, R. J. (1980). Approximation theorems of mathematical statistics. Wiley.
• Sheng, W., & Yin, X. (2013). Direction estimation in single-index models via distance covariance. Journal of Multivariate Analysis, 122, 148–161. https://doi.org/10.1016/j.jmva.2013.07.003 
• Sheng, W., & Yin, X. (2016). Sufficient dimension reduction via distance covariance. Journal of Computational and Graphical Statistics, 25(1), 91–104. https://doi.org/10.1080/10618600.2015.1026601 
• Sheng, W., & Yuan, Q. (2020). Sufficient dimension folding in regression via distance covariance for matrix-valued predictors. Statistical Analysis and Data Mining, 13(1), 71–82. https://doi.org/10.1002/sam.v13.1 
• Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267–288. https://doi.org/10.1111/rssb.1996.58.issue-1
• Waltz, R. A., Morales, J. L., & Orban, D. (2006). An interior algorithm for nonlinear optimization that combines line search and trust region steps. Mathematical Programming, 107(3), 391–408. https://doi.org/10.1007/s10107-004-0560-5 
• Wang, H., & Xia, Y. (2008). Sliced regression for dimension reduction. Journal of the American Statistical Association, 103(482), 811–821. https://doi.org/10.1198/016214508000000418 
• Wang, Q., Yin, X., & Critchley, F. (2015). Dimension reduction based on the hellinger integral. Biometrika, 102(1), 95–106. https://doi.org/10.1093/biomet/asu062
• Xia, Y. (2007). A constructive approach to the estimation of dimension reduction directions. The Annals of Statistics, 35(6), 2654–2690. https://doi.org/10.1214/009053607000000352
• Xia, Y., Tong, H., Li, W. K., & Zhu, L.-X. (2002). An adaptive estimation of dimension reduction space. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 64(3), 363–410. https://doi.org/10.1111/rssb.2002.64.issue-3 
• Ye, Z., & 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(464), 968–979. https://doi.org/10.1198/016214503000000927 
• Yin, X., & Hilafu, H. (2015). Sequential sufficient dimension reduction for large p, small n problems. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(4), 879–892. https://doi.org/10.1111/rssb.2015.77.issue-4 
• Yin, X., Li, B., & Cook, R. D. (2008). Successive direction extraction for estimating the central subspace in a multiple-index regression. Journal of Multivariate Analysis, 99(8), 1733–1757. https://doi.org/10.1016/j.jmva.2008.01.006
• Yin, X., & Yuan, Q. (2020). A new class of measures for testing independence. Statistica Sinica, 30(4), 2131–2154. 
• Zeng, P., & Zhu, Y. (2010). An integral transform method for estimating the central mean and central subspace. Journal of Multivariate Analysis, 101(1), 271–290. https://doi.org/10.1016/j.jmva.2009.08.004
• Zhu, L., & Fang, K. (1996). Asymptotics for kernel estimate of sliced inverse regression. The Annals of Statistics, 24(3), 1053–1068. https://doi.org/10.1214/aos/1032526955
• Zhu, Y., & Zeng, P. (2006). Fourier methods for estimating the central subspace and the central mean subspace in regression. Journal of the American Statistical Association, 101(476), 1638–1651. https://doi.org/10.1198/016214506000000140 

To cite this article: Wenhui Sheng & Qingcong Yuan (2023) Dimension reduction with expectation of conditional difference measure, Statistical Theory and Related Fields, 7:3, 188-201, DOI: 10.1080/24754269.2023.2182136

To link to this article: https://doi.org/10.1080/24754269.2023.2182136