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

Author

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  • Eric Gautier

    () (TSE - Toulouse School of Economics - Toulouse School of Economics)

  • Erwan Le Pennec

    () (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique, XPOP - Modélisation en pharmacologie de population - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en Automatique)

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.

Suggested Citation

  • Eric Gautier & Erwan Le Pennec, 2017. "Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding," Working Papers inria-00601274, HAL.
  • Handle: RePEc:hal:wpaper:inria-00601274
    Note: View the original document on HAL open archive server: https://hal.inria.fr/inria-00601274v4
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    References listed on IDEAS

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    1. Heckman, James & Singer, Burton, 1984. "A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data," Econometrica, Econometric Society, vol. 52(2), pages 271-320, March.
    2. Hoderlein, Stefan & Klemelä, Jussi & Mammen, Enno, 2010. "Analyzing The Random Coefficient Model Nonparametrically," Econometric Theory, Cambridge University Press, vol. 26(03), pages 804-837, June.
    3. Eric Gautier & Stefan Soderlein, 2011. "Estimating the Distribution of Treatment Effects," Working Papers 2011-25, Center for Research in Economics and Statistics.
    4. Ichimura, Hidehiko & Thompson, T. Scott, 1998. "Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution," Journal of Econometrics, Elsevier, vol. 86(2), pages 269-295, June.
    5. Eric Gautier & Yuichi Kitamura, 2013. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Econometrica, Econometric Society, vol. 81(2), pages 581-607, March.
    6. Chris Elbers & Geert Ridder, 1982. "True and Spurious Duration Dependence: The Identifiability of the Proportional Hazard Model," Review of Economic Studies, Oxford University Press, vol. 49(3), pages 403-409.
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    Citations

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    Cited by:

    1. Christoph Breunig, 2018. "Varying Random Coefficient Models," Papers 1804.03110, arXiv.org.
    2. Durastanti, Claudio, 2016. "Adaptive global thresholding on the sphere," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 110-132.
    3. Fabian Dunker & Stefan Hoderlein & Hiroaki Kaido, 2013. "Random Coefficients in Static Games of Complete Information," Boston College Working Papers in Economics 835, Boston College Department of Economics.
    4. Gautier, Eric & Hoderlein, Stefan, 2011. "A triangular treatment effect model with random coefficients in the selection equation," TSE Working Papers 15-598, Toulouse School of Economics (TSE), revised 25 Aug 2015.
    5. Andrew Chesher & Adam M. Rosen, 2014. "An instrumental variable random‐coefficients model for binary outcomes," Econometrics Journal, Royal Economic Society, vol. 17(2), pages 1-19, June.
    6. Xiaohong Chen & Timothy M. Christensen, 2015. "Optimal sup-norm rates, adaptivity and inference in nonparametric instrumental variables estimation," CeMMAP working papers CWP32/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Xiaohong Chen & Timothy Christensen, 2013. "Optimal Sup-norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation," Cowles Foundation Discussion Papers 1923R, Cowles Foundation for Research in Economics, Yale University, revised Apr 2015.

    More about this item

    Keywords

    adaptation; inverse problems; data-driven thresholding; random coefficients; minimax rate optimality; needlets; Discrete choice models;

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