An Adaptive Estimation of Dimension Reduction Space

Abstract

Summary. Searching for an effective dimension reduction space is an important problem in regression, especially for high dimensional data. We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.

Group-wise sufficient dimension reduction with principal fitted components

“…The second stage is to evaluate the models under the different structures of induced by the clustering. In our investigation, we have considered developing a similar procedure for the following methods: SIR of Li (1991), SAVE of Cook and Weisberg (1991), LAD of Cook and Forzani (2009), and MAVE of Xia et al (2002). However, major issues arose.…”

“…The central subspace was also estimated by LAD using the package ldr of Adragni and Raim (2014a). We implemented the algorithm of MAVE following Xia et al (2002). Figure 7 shows the angle in degrees between the true and the estimated central subspaces, both of dimension one in R 15 .…”

“…Some recently developed methods, such as principal fitted components (PFC; Cook 2007) and likelihood acquired directions (LAD; Cook and Forzani 2009) are likelihood-based. Kernel-based methods such as the minimum average variance estimation (MAVE; Xia et al 2002) and its variants have also been proposed.…”

Robust estimation for varying index coefficient models

“…The VICM is an important tool in multivariate semiparametric regression, which is an extension of the varying coefficient model (Hastie and Tibshirani 1993;Cai et al 2000;Fan and Zhang 2008), single-index models (Xia et al 2002;Cui et al 2011), among others. On the one hand, various existing semiparametric models are special cases of varying index coefficient models.…”

Dimension Reduction for Gaussian Process Emulation: An Application to the Influence of Bathymetry on Tsunami Heights

“…Several methods have been developed to find SDR, including nonparametric approaches such as sliced inverse regression (SIR) [24] and minimum average variance estimation (MAVE) [35], and parametric approaches like principal fitted components (PFC) [8,10]. In this paper, we adopt the gradient-based kernel dimension reduction (gKDR) [15] to construct low-dimensional approximations to the simulators.…”