An adaptive estimation of dimension reduction space

Yingcun Xia, Howell Tong*, W. K. Li, Lixing ZHU

*Corresponding author for this work

Research output: Chapter in book/report/conference proceedingChapterpeer-review

7 Citations (Scopus)


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.

Original languageEnglish
Title of host publicationExploration of a Nonlinear World
Subtitle of host publicationAn Appreciation of Howell Tong's Contributions to Statistics
PublisherWorld Scientific Publishing Co.
Number of pages48
ISBN (Electronic)9789812836281
ISBN (Print)9812836276, 9789812836274
Publication statusPublished - 1 Jan 2009

Scopus Subject Areas

  • Business, Management and Accounting(all)
  • Economics, Econometrics and Finance(all)
  • Mathematics(all)

User-Defined 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


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