Dimension Reduction in Regressions Through Cumulative Slicing Estimation

Abstract

In this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction. In parallel to the classical slicing estimation, we develop three methods that are termed, respectively, as cumulative mean estimation, cumulative variance estimation, and cumulative directional regression. The strong consistency for p = O(n 1/2 / log n) and the asymptotic normality for p = o(n 1/2 ) are established, where p is the dimension of the predictors and n is sample size. Such asymptotic results improve the rate p = o(n 1/3 ) in many existing contexts of semiparametric modeling. In addition, we propose a modified BIC-type criterion to estimate the structural dimension of the central subspace. Its consistency is established when p = o(n 1/2 ). Extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration.

A new sliced inverse regression method for multivariate response

“…For instance, it is straightforward to build multivariate regularized SIR approaches from Bernard-Michel et al (2009b), multivariate SIR α approaches from Gannoun and Saracco (2003) or multivariate kernel SIR methods from Wu (2008) and to obtain the associated clustering step. To avoid the choice of the number H of slices, one may also consider building a multivariate version of the CUME procedure from Zhu et al (2010a).…”

Probability-enhanced effective dimension reduction for classifying sparse functional data

“…Recently, Jiang et al (2014) proposed an inverse regression method for sparse functional data by estimating the inverse conditional mean functions with a two-dimensional smoother that requires considerable computation. Inspired by cumulative slicing estimation (CUME) for multivariate data (Zhu et al 2010), Yao et al (2015) proposed to borrow information across subjects via a one-dimensional smoother, named the functional cumulative slicing (FCS), which is closely related to the proposed method in this paper. The EDR methods are intended for continuous response and have been rarely used for classification problems due to few distinct response values.…”

“…Thus, one major advantage of EDR methods is "link-free" (Duan and Li 1991). Pioneered by Li (1991) that proposed the sliced inverse regression (SIR) using the information concerning the inverse conditional mean E(X |Y ), Cook and Weisberg (1991) considered the inverse variance estimation utilizing the information of var(X |Y ), Li (1992) dealt with the Hessian matrix of the regression curve, Chiaromonte et al (2002) modified sliced inverse regression for categorical predictors, Li and Wang (2007) worked with empirical directions, and Zhu et al (2010) proposed cumulative slicing estimation to improve upon SIR.…”

“…To avoid the nontrivial selection of the number of slices in multivariate sliced inverse regression, Zhu et al (2010) observed that for a fixedỹ, using two slices I 1 = (−∞,ỹ] and I 2 = (ỹ, +∞) would maximize the use of data and minimize the variability in each slice. Viewing the limitation of the functional SIR for sparse functional data, the choice of cumulative slicing is thus critical to ensure sufficient number of observations.…”

An Adaptive-to-Model Test for Parametric Single-Index Errors-in-Variables Models

“…There are a number of proposals available in the literature for this purpose. Examples include sliced inverse regression (SIR) of Li (1991), sliced average variance estimation (SAVE) of Cook & Weisberg (1991), minimum average variance estimation (MAVE) of Xia et al (2002), contour regression (CR) of Li et al (2005), directional regression (DR) of Li & Wang (2007), discretization-expectation estimation (DEE) of Zhu et al (2010a), and the average partial mean estimation (APME) of Zhu et al (2010b). When there are measurement errors, the above methods need a modification for consistently estimating the matrix B up to a q × q orthonormal matrix C.…”