## The Annals of Statistics

- Ann. Statist.
- Volume 19, Number 2 (1991), 505-530.

### Slicing Regression: A Link-Free Regression Method

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 19, Number 2 (1991), 505-530.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176348109

**Digital Object Identifier**

doi:10.1214/aos/1176348109

**Mathematical Reviews number (MathSciNet)**

MR1105834

**Zentralblatt MATH identifier**

0738.62070

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J99: None of the above, but in this section

**Keywords**

Elliptical symmetry general regression model inverse regression projection pursuit spectral decomposition

#### Citation

Duan, Naihua; Li, Ker-Chau. Slicing Regression: A Link-Free Regression Method. Ann. Statist. 19 (1991), no. 2, 505--530. doi:10.1214/aos/1176348109. https://projecteuclid.org/euclid.aos/1176348109