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February 2022 Adaptive estimation in the linear random coefficients model when regressors have limited variation
Christophe Gaillac, Eric Gautier
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Bernoulli 28(1): 504-524 (February 2022). DOI: 10.3150/21-BEJ1354


We consider a linear model where the coefficients – intercept and slopes – are random and independent from the regressors. The law of the coefficients is nonparametric. Without further restriction, nonparametric identification requires the regressors to have a support which is the whole space. This is hardly ever the case in practice. It is possible to handle regressors with limited variation when the coefficients can have a compact support. This is not compatible with unbounded error terms as usual in regression models. In this paper, the regressors can have a support which is a proper subset but the slopes do not have heavy-tails. Lower bounds on the minimax risk for the estimation of the joint density of the random coefficients density are obtained for a wide range of smoothness. Some allow for polynomial and nearly parametric rates of convergence. We present a minimax optimal estimator and a data-driven rule for adaptive estimation. A R package is available to implement this estimator.


The authors acknowledge financial support from the grants ERC POEMH 337665 and ANR-17-EURE-0010. Christophe Gaillac thanks CREST/ENSAE where this research was partly conducted. The authors are grateful to the seminar participants at Berkeley, Brown, CREST, Duke, Harvard-MIT, Rice, TSE, ULB, University of Tokyo, those of 2016 SFDS, ISNPS, Recent Advances in Econometrics, and 2017 IAAE conferences for comments.


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Christophe Gaillac. Eric Gautier. "Adaptive estimation in the linear random coefficients model when regressors have limited variation." Bernoulli 28 (1) 504 - 524, February 2022.


Received: 1 July 2020; Revised: 1 April 2021; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1354

Keywords: Adaptation , inverse problem , minimax , random coefficients

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 1 • February 2022
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