## Electronic Journal of Statistics

### 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 $\mathrm{L}^{p}(\mathbb{S}^{d-1})$ 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.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 277-320.

Dates
First available in Project Euclid: 12 February 2018

https://projecteuclid.org/euclid.ejs/1518426111

Digital Object Identifier
doi:10.1214/17-EJS1383

Mathematical Reviews number (MathSciNet)
MR3763073

Zentralblatt MATH identifier
1387.62049

Subjects
Secondary: 42C15, 62C20, 62G07, 62G08, 62G20

#### Citation

Gautier, Eric; Le Pennec, Erwan. Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding. Electron. J. Statist. 12 (2018), no. 1, 277--320. doi:10.1214/17-EJS1383. https://projecteuclid.org/euclid.ejs/1518426111

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