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June, 1991 Slicing Regression: A Link-Free Regression Method
Naihua Duan, Ker-Chau Li
Ann. Statist. 19(2): 505-530 (June, 1991). DOI: 10.1214/aos/1176348109


Consider a general regression model of the form $y = g(\alpha + \mathbf{x}'\beta, \varepsilon)$, with an arbitrary and unknown link function $g$. We study a link-free method, the slicing regression, for estimating the direction of $\beta$. The method is easy to implement and does not require iterative computation. First, we estimate the inverse regression function $E(\mathbf{x}\mid y)$ using a step function. We then estimate $\Gamma = \operatorname{Cov}\lbrack E(\mathbf{x}\mid y)\rbrack$, using the estimated inverse regression function. Finally, we take the spectral decomposition of the estimate $\hat\Gamma$ with respect to the sample covariance matrix for $\mathbf{x}$. The principal eigenvector is the slicing regression estimate for the direction of $\beta$. We establish $\sqrt n$-consistency and asymptotic normality, derive the asymptotic covariance matrix and provide Wald's test and a confidence region procedure. Efficiency is discussed for an important special case. Most of our results require $\mathbf{x}$ to have an elliptically symmetric distribution. When the elliptical symmetry is violated, a bias bound is provided; the asymptotic bias is small when the elliptical symmetry is nearly satisfied. The bound suggests a projection index which can be used to measure the deviation from elliptical symmetry. The theory is illustrated with a simulation study.


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Naihua Duan. Ker-Chau Li. "Slicing Regression: A Link-Free Regression Method." Ann. Statist. 19 (2) 505 - 530, June, 1991.


Published: June, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0738.62070
MathSciNet: MR1105834
Digital Object Identifier: 10.1214/aos/1176348109

Primary: 62J99

Keywords: Elliptical symmetry , general regression model , inverse regression , Projection pursuit , spectral decomposition

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • June, 1991
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