  We propose a novel approach to sufficient dimension reduction in regression,
based on estimating contour directions of negligible variation for the response
surface. These directions span the orthogonal complement of the minimal space
relevant for the regression, and can be extracted according to a measure of the
variation in the response, leading to General Contour Regression(GCR). In
comparison to exiisting sufficient dimension reduction techniques, this
sontour-based mothology guarantees exhaustive estimation of the central space
under ellipticity of the predictoor distribution and very mild additional
assumptions, while maintaining vn-consisytency and somputational ease.
Moreover, it proves to be robust to departures from ellipticity. We also
establish some useful population properties for GCR. Simulations to compare
performance with that of standard techniques such as ordinary least squares,
sliced inverse regression, principal hessian directions, and sliced average
variance estimation confirm the advntages anticipated by theoretical analyses.
We also demonstrate the use of contour-based methods on a data set concerning
grades of students from Massachusetts colleges.
