  In the random coefficients binary choice model, a binary variable equals 1
iff an index $X^\top\beta$ is positive.The vectors $X$ and $\beta$ are
independent and belong to the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$.We
prove lower bounds on the minimax risk for estimation of the density
$f\_{\beta}$ over Besov bodies where the loss is a power of the
$L^p(\mathbb{S}^{d-1})$ norm for $1\le p\le \infty$. We show that a hard
thresholding estimator based on a needlet expansion with data-driven thresholds
achieves these lower bounds up to logarithmic factors.
