An adaptive estimation of dimension reduction space

Yingcun XIA, Howell TONG, Wai Keung LI, Li-Xing ZHU

Research output: Chapter in Book/Report/Conference proceedingChapter


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 under smooth 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. Copyright © 2002 Royal Statistical Society.
Original languageEnglish
Title of host publicationExploration of a nonlinear world: An appreciation of Howell Tong's contributions to statistics
EditorsKung-Sik CHAN
Place of PublicationSingapore
PublisherWorld Scientific
ISBN (Electronic)9789812836281
ISBN (Print)9789812836274
Publication statusPublished - 2009


Adaptive Estimation
Dimension Reduction
Variance Estimation
Link Function
High-dimensional Data
Effective Dimension
Semiparametric Model
Time series


Xia, Y., Tong, H., Li, W. K., & Zhu, L.-X. (2009). An adaptive estimation of dimension reduction space. In K.-S. Chan (Ed.), Exploration of a nonlinear world: An appreciation of Howell Tong's contributions to statistics (pp. 299-346). Singapore: World Scientific.