J. R. Statist. Soc. B (2002) 64, Part 3, pp. 363–410
An adaptive estimation of dimension reduction space
Yingcun Xia, University of Cambridge, UK, and Jinan University, People’s Republic of China
Howell Tong, University of Hong Kong, People’s Republic of China, and London School of Economics and Political Science, UK
W. K. Li University of Hong Kong, People’s Republic of China
and Li-Xing Zhu University of Hong Kong and Chinese Academy of Sciences, Beijing, People’s Republic of China
[Read before The Royal Statistical Society at a meeting organized by the Research Section on Wednesday, February 13th, 2002, Professor D. Firth in the Chair]
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 with- out 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 theo- retical 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. Keywords: Average derivative estimation; Dimension reduction; Generalized linear models; Local linear smoother; Multiple time series; Non-linear time series analysis; Nonparametric regression; Principal Hessian direction; Projection pursuit; Semiparametrics; Sliced inverse regression estimation
1. Introduction Let y and X be respectively R-valued and Rp-valued random variables. Without prior knowledge about the relationship between y and X, the regression function g.x/ = E.y|X = x/ is often
Address for correspondence: Howell Tong, Department of Statistics, London School of Economics and Political Science, Houghton Street, London, WC2A 2AE, UK. E-mail: [email protected]
2002 Royal Statistical Society 1369–7412/02/64363 364 Y. Xia, H. Tong, W. K. Li and L.-X. Zhu modelled in a flexible nonparametric fashion. When the dimension of X is high, recent efforts have been expended in finding the relationship between y and X efficiently. The final goal is to approximate g.x/ by a function having simplifying structure which makes estimation and interpretation possible even for moderate sample sizes. There are essentially two approaches: the first is largely concerned with function approximation and the second with dimension reduction. Examples of the former are the additive model approach of Hastie and Tibshirani (1986) and the projection pursuit regression proposed by Friedman and Stuetzle (1981); both assume that the regression function is a sum of univariate smooth functions. Examples of the latter are the dimension reduction of Li (1991) and the regression graphics of Cook (1998). A regression-type model for dimension reduction can be written as
T y = g.B0 X/ + "; (1.1) where g is an unknown smooth link function, B0 = .β1;:::;βD/ is a p × D orthogonal matrix T (B0 B0 = ID×D) with D