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Optimal sufficient dimension reduction in regressions with categorical predictors. (English) Zbl 1118.62043
Summary: Though partial sliced inverse regression [partial SIR; F. Chiaromonte et al., Sufficient dimension reduction in regressions with categorical predictors. Ann. Statist. 30, No. 2, 475–497 (2002; Zbl 1012.62036)] extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors, its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice. When this condition fails, partial SIR may provide misleading results.
We propose a new estimation method via a minimum discrepancy approach without this restriction. Our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi-squared distribution. It also gives us the ability to test predictor effects. An asymptotic chi-squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62H10 Multivariate distribution of statistics
62E20 Asymptotic distribution theory in statistics
62H15 Hypothesis testing in multivariate analysis
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[1] Browne, M.W., Asymptotic distribution-free methods for the analysis of covariance structures, British J. math. statist. psychology, 37, 62-83, (1984) · Zbl 0561.62054
[2] Carroll, R.; Ruppert, D., Transformation and weighting in regression, (1988), Chapman & Hall London · Zbl 0666.62062
[3] Chiaromonte, F.; Cook, R.D., Sufficient dimension reduction and graphics in regression, Ann. inst. statist. math., 54, 768-795, (2002) · Zbl 1047.62066
[4] Chiaromonte, F.; Cook, R.D.; Li, B., Sufficient dimension reduction in regressions with categorical predictors, Ann. statist., 30, 475-497, (2002) · Zbl 1012.62036
[5] Cook, R.D., Using dimension-reduction subspaces to identify important inputs in models of physical systems, (), 18-25
[6] Cook, R.D., Regression graphics, (1998), Wiley New York, NY · Zbl 0903.62001
[7] Cook, R.D., SAVE: a method for dimension reduction and graphics in regression, Comm. statist.: theory methods, 29, 161-175, (2000)
[8] Cook, R.D., Testing predictor contributions in sufficient dimension reduction, Ann. statist., 32, 1062-1092, (2004) · Zbl 1092.62046
[9] Cook, R.D.; Nachtsheim, C.J., Reweighting to achieve elliptically contoured covariates in regression, J. amer. statist. assoc., 89, 592-599, (1994) · Zbl 0799.62078
[10] Cook, R.D.; Ni, L., Sufficient dimension reduction via inverse regression: a minimum discrepancy approach, J. amer. statist. assoc., 100, 410-428, (2005) · Zbl 1117.62312
[11] Cook, R.D.; Weisberg, S., Discussion of sliced inverse regression for dimension reduction, J. amer. statist. assoc., 86, 316-342, (1991)
[12] Cook, R.D.; Weisberg, S., Partial one-dimensional regression models, Amer. statist., 58, 110-116, (2004)
[13] Ferguson, T., A method of generating best asymptotically normal estimates with application to the estimation of bacterial densities, Ann. math. statist., 29, 1046-1062, (1958) · Zbl 0089.15402
[14] Friedman, J.H., An overview of predictive learning and function approximation, from statistics to neural networks, ()
[15] Hall, P.; Li, K.C., On almost linearity of low dimensional projections from high dimensional data, Ann. statist., 21, 867-889, (1993) · Zbl 0782.62065
[16] Li, K.C., Sliced inverse regression for dimension reduction (with discussion), J. amer. statist. assoc., 86, 316-342, (1991)
[17] Li, B.; Cook, R.D.; Chiaromonte, F., Dimension reduction for the conditional Mean in regressions with categorical predictors, Ann. statist., 31, 1636-1668, (2003) · Zbl 1042.62037
[18] Lindsay, B.; Qu, A., Inference functions and quadratic score tests, Statist. sci., 18, 394-410, (2003) · Zbl 1055.62047
[19] Ruhe, A.; Wedin, P.A., Algorithms for separable nonlinear least squares problems, SIAM review,, 22, 318-337, (1980) · Zbl 0466.65039
[20] Shapiro, A., Asymptotic theory of overparameterized structural model, J. amer. statist. assoc., 81, 142-149, (1986) · Zbl 0596.62069
[21] Velilla, S., Assessing the number of linear components in a general regression problem, J. amer. statist. assoc., 93, 1088-1098, (1998) · Zbl 1063.62553
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