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

Abstract

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.

References in corpus (5)

• Empirical Bernstein Bounds and Sample Variance Penalization
• Periodic boxcar deconvolution and diophantine approximation
• Wavelet regression in random design with heteroscedastic dependent errors
• Needlet algorithms for estimation in inverse problems
• Deconvolution in white noise with a random blurring function