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

Asymptotic normality of a linear threshold estimator in fixed dimension with near-optimal rate