A constructive approach to the estimation of dimension reduction directions

By Yingcun Xia. 2007

208 citations recorded by the source corpus.

In this paper we propose two new methods to estimate the dimension-reduction directions of the central subspace (CS) by constructing a regression model such that the directions are all captured in the regression mean. Compared with the inverse regression estimation methods [e.g., J. Amer. Statist. Assoc. 86 (1991) 328-332, J. Amer. Statist. Assoc. 86 (1991) 316-342, J. Amer. Statist. Assoc. 87 (1992) 1025-1039], the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable, and have better performance than the inverse regression estimation methods for finite samples. Compared with the direct regression estimation methods [e.g., J. Amer. Statist. Assoc. 84 (1989) 986-995, Ann. Statist. 29 (2001) 1537-1566, J. R. Stat. Soc. Ser. B Stat. Methodol. 64 (2002) 363-410], which can only estimate the directions of CS in the regression mean, the new methods can detect the directions of CS exhaustively. Consistency of the estimators and the convergence of corresponding algorithms are proved.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/856c9759c48991f4ebdbac6ea5f4e04ee218d7f6

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In Management Science and Operations Research

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Citations in the corpus

Successive direction extraction for estimating the central subspace in a multiple-index regression (2008, methodology): “A recent work of Xia (2007), which combines the kernel conditional density estimation with OPG and MAVE, also recovers the CS. Xia’s method differs from ours in that it treats the predictor X jointly, and therefore involves a multivariate kernel estimate for the density of X.”
On estimation efficiency of the central mean subspace (2014, methodology): “See for example the sequential test procedure in Cook and Li (2004), the crossvalidation procedure in Xia et al. (2002) and Xia (2007), the Bayesian information criterion BIC type of criterion in Zhu et al. (2006) and the bootstrap procedure in Ye and Weiss (2003), Li and Dong (2009), Dong and Li…”
Groupwise Dimension Reduction (2010, methodology): “The above conditions are the same as part of those designed in Xia et al. (2002), but their conditions on the kernel is unnecessary in our setup because of the Gaussian kernel we use.”
Sufficient Dimension Reduction via Squared-Loss Mutual Information Estimation (2010, methodology): “• Minimum Average (conditional) Variance Estimation (dMAVE) (Xia, 2007).”
Model checking for parametric single‐index models: a dimension reduction model‐adaptive approach (2016, methodology): “Recently, Xia (2007) analysed this data set with his method.”
A Simple Two-Sample Test in High Dimensions Based on L2-Norm (2020, background): “In regression, feature screening and selection, and dimension reduction are the mainstream approaches (Cheng and Wu 2013; Cook and Li 2002; Fan and Li 2001; Fan and Lv 2008; Fan and Peng 2004; Fan and Song 2010; Lv and Fan 2009; Li et al. 2012; Tibshirani 1996; Xia 2007, 2008).”
Sufficient dimension reduction based on an ensemble of minimum average variance estimators (2011, methodology): “Besides MAVE, Xia et al. (2002) also introduced two companion estimators: the outer product of gradients (OPG) and a refinement of MAVE (RMAVE).”
A Multiple-Index Model and Dimension Reduction (2008, methodology): “However, the estimation combining the two ideas is not as efficient as the method introduced in Xia et al. (2002); see the discussion in Xia (2006).”
Functional single index models for longitudinal data (2011, unknown): “Theoretical results for this approach to single-index models are available in Xia (2006) and some extensions have been studied in Xia (2007) and Kong and Xia (2007), among others.”