Adaptive estimation in the linear random coefficients model when regressors have limited variation

Christophe Gaillac; Eric Gautier

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arXiv · 1905.06584

Adaptive estimation in the linear random coefficients model when regressors have limited variation

Christophe Gaillac·Eric Gautier

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Abstract

We consider a linear model where the coefficients - intercept and slopes - are random with a law in a nonparametric class and independent from the regressors. Identification often requires the regressors to have a support which is the whole space. This is hardly ever the case in practice. Alternatively, the coefficients can have a compact support but 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 (not the intercept) do not have heavy-tails. Lower bounds on the supremum risk for the estimation of the joint density of the random coefficients density are obtained for a wide range of smoothness, where some allow for polynomial and nearly parametric rates of convergence. We present a minimax optimal estimator, a data-driven rule for adaptive estimation, and made available a R package.

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Christophe Gaillac, Eric Gautier. 2019-05-16. Adaptive estimation in the linear random coefficients model when regressors have limited variation. https://arxiv.org/abs/1905.06584

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