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December 2004 Determining the dimension of iterative Hessian transformation
R. Dennis Cook, Bing Li
Ann. Statist. 32(6): 2501-2531 (December 2004). DOI: 10.1214/009053604000000661


The central mean subspace (CMS) and iterative Hessian transformation (IHT) have been introduced recently for dimension reduction when the conditional mean is of interest. Suppose that X is a vector-valued predictor and Y is a scalar response. The basic problem is to find a lower-dimensional predictor ηTX such that E(Y|X)=E(YTX). The CMS defines the inferential object for this problem and IHT provides an estimating procedure. Compared with other methods, IHT requires fewer assumptions and has been shown to perform well when the additional assumptions required by those methods fail. In this paper we give an asymptotic analysis of IHT and provide stepwise asymptotic hypothesis tests to determine the dimension of the CMS, as estimated by IHT. Here, the original IHT method has been modified to be invariant under location and scale transformations. To provide empirical support for our asymptotic results, we will present a series of simulation studies. These agree well with the theory. The method is applied to analyze an ozone data set.


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R. Dennis Cook. Bing Li. "Determining the dimension of iterative Hessian transformation." Ann. Statist. 32 (6) 2501 - 2531, December 2004.


Published: December 2004
First available in Project Euclid: 7 February 2005

zbMATH: 1069.62033
MathSciNet: MR2153993
Digital Object Identifier: 10.1214/009053604000000661

Primary: 62G08
Secondary: 62G09 , 62H05

Keywords: asymptotic test , conditional mean , Dimension reduction , Eigenvalues , order determination

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 6 • December 2004
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