Section: New Results

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 $X^T\beta$ is positive. The vectors $X$ and $\beta$ are independent and belong to the sphere $S^{d-1}$ in $R^d$. We have proven lower bounds on the minimax risk for estimation of the density $f_{\beta }$ over Besov bodies where the loss is a power of the $L^p\left(S^{d-1}\right)$ norm for $1\le p\le \infty$. We have shown that a hard thresholding estimator based on a needlet expansion with data-driven thresholds achieves these lower bounds up to logarithmic factors.