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 $X^{\mathrm{\top }}\beta$ 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 $f\mathrm{\_}\beta$ over Besov bodies where the loss is a power of the $L^p({\mathbb{S}}^{d-1})$ norm for $1\le p\le \mathrm{\infty }$. We show that a hard thresholding estimator based on a needlet expansion with data-driven thresholds achieves these lower bounds up to logarithmic factors.