  We propose a novel approach to sufficient dimension reduction in regression,
based on estimating contour directions of small variation in the response.
These directions span the orthogonal complement of the minimal space relevant
for the regression and can be extracted according to two measures of variation
in the response, leading to simple and general contour regression (SCR and GCR)
methodology. In comparison with existing sufficient dimension reduction
techniques, this contour-based methodology guarantees exhaustive estimation of
the central subspace under ellipticity of the predictor distribution and mild
additional assumptions, while maintaining \sqrtn-consistency and computational
ease. Moreover, it proves robust to departures from ellipticity. We establish
population properties for both SCR and GCR, and asymptotic properties for SCR.
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 advantages anticipated by
the theoretical analyses. We demonstrate the use of contour-based methods on a
data set concerning soil evaporation.
