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.