## The Annals of Statistics

### Determining the dimension of iterative Hessian transformation

#### Abstract

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.

#### Article information

Source
Ann. Statist. Volume 32, Number 6 (2004), 2501-2531.

Dates
First available in Project Euclid: 7 February 2005

http://projecteuclid.org/euclid.aos/1107794877

Digital Object Identifier
doi:10.1214/009053604000000661

Mathematical Reviews number (MathSciNet)
MR2153993

Zentralblatt MATH identifier
1069.62033

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G09: Resampling methods 62H05: Characterization and structure theory

#### Citation

Cook, R. Dennis; Li, Bing. Determining the dimension of iterative Hessian transformation. Ann. Statist. 32 (2004), no. 6, 2501--2531. doi:10.1214/009053604000000661. http://projecteuclid.org/euclid.aos/1107794877.

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