  Linear thresholding models postulate that the conditional distribution of a
response variable in terms of covariates differs on the two sides of a
(typically unknown) hyperplane in the covariate space. A key goal in such
models is to learn about this separating hyperplane. Exact likelihood or least
squares methods to estimate the thresholding parameter involve an indicator
function which make them difficult to optimize and are, therefore, often
tackled by using a surrogate loss that uses a smooth approximation to the
indicator. In this paper, we demonstrate that the resulting estimator is
asymptotically normal with a near optimal rate of convergence: $n^{-1}$ up to a
log factor, in both classification and regression thresholding models. This is
substantially faster than the currently established convergence rates of
smoothed estimators for similar models in the statistics and econometrics
literatures. We also present a real-data application of our approach to an
environmental data set where $CO_2$ emission is explained in terms of a
separating hyperplane defined through per-capita GDP and urban agglomeration.
