  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.
