Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding

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.