Open Access
2023 Asymptotic normality of a change plane estimator in fixed dimension with near-optimal rate
Debarghya Mukherjee, Moulinath Banerjee, Debasri Mukherjee, Ya’acov Ritov
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Electron. J. Statist. 17(2): 2289-2316 (2023). DOI: 10.1214/23-EJS2144


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: n1 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 CO2 emission is explained in terms of a separating hyperplane defined through per-capita GDP and urban agglomeration.


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Debarghya Mukherjee. Moulinath Banerjee. Debasri Mukherjee. Ya’acov Ritov. "Asymptotic normality of a change plane estimator in fixed dimension with near-optimal rate." Electron. J. Statist. 17 (2) 2289 - 2316, 2023.


Received: 1 March 2022; Published: 2023
First available in Project Euclid: 4 October 2023

MathSciNet: MR4649982
Digital Object Identifier: 10.1214/23-EJS2144

Keywords: Change plane estimator , CO2 emission data , near-optimal rate

Vol.17 • No. 2 • 2023
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