  We consider the problem of estimating the region on which a non-parametric
regression function is at its baseline level in two dimensions. The baseline
level typically corresponds to the minimum/maximum of the function and
estimating such regions or their complements is pertinent to several problems
arising in edge estimation, environmental statistics, fMRI and related fields.
We assume the baseline region to be convex and estimate it via fitting a
`stump' function to approximate $p$-values obtained from tests for deviation of
the regression function from its baseline level. The estimates, obtained using
an algorithm originally developed for constructing convex contours of a
density, are studied in two different sampling settings, one where several
responses can be obtained at a number of different covariate-levels
(dose-response) and the other involving limited number of response values per
covariate (standard regression). The shape of the baseline region and the
smoothness of the regression function at its boundary play a critical role in
determining the rate of convergence of our estimate: for a regression function
which is `p-regular' at the boundary of the convex baseline region, our
estimate converges at a rate $N^{2/(4p+3)}$ in the dose-response setting, $N$
being the total budget, and its analogue in the standard regression setting
converges at a rate of $N^{1/(2p+2)}$. Extensions to non-convex baseline
regions are explored as well.
