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^Tβ is positive. The vectors X and β are independent and belong to the sphere S^d−1 in Rd. We prove lower bounds on the minimax risk for estimation of the density fβ over Besov bodies where the loss is a power of the L^p(S^d−1) norm for 1 ≤ p ≤ ∞. We show that a hard thresholding estimator based on a needlet expansion with data-driven thresholds achieves these lower bounds up to logarithmic factors.