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Consider a general regression model of the form y = g (+ x', ), with an arbitrary and unknown link function g. We study a link-free method, the slicing regression, for estimating the direction of. The method is easy to implement and does not require iterative computation. First, we estimate the inverse regression function E (x y) using a step function. We then estimate = Cov E (x y), using the estimated inverse regression function. Finally, we take the spectral decomposition of the estimate with respect to the sample covariance matrix for x. The principal eigenvector is the slicing regression estimate for the direction of. We establish 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 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.
Duan et al. (Sat,) studied this question.