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

TL;DR

This paper develops an adaptive needlet thresholding method for estimating the density of random coefficients in a binary choice model, achieving near-optimal minimax risk bounds over Besov spaces.

Contribution

It introduces a novel needlet-based estimator with data-driven thresholds that attains minimax optimal rates in the nonparametric binary choice model.

Findings

Estimator achieves minimax lower bounds up to logarithmic factors.

Method adapts to unknown smoothness of the density.

Provides theoretical guarantees for estimation accuracy.

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.