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

  • Eric Gautier


    (CREST - Centre de Recherche en Économie et Statistique - INSEE - École Nationale de la Statistique et de l'Administration Économique, ENSAE - École Nationale de la Statistique et de l'Administration Économique - ENSAE ParisTech)

  • Erwan Le Pennec


    (SELECT - Model selection in statistical learning - LMO - Laboratoire de Mathématiques d'Orsay - Université Paris Saclay - INRIA Saclay - Ile de France - INRIA - CNRS, LM-Orsay - Laboratoire de Mathématiques d'Orsay - CNRS - UP11 - Université Paris-Sud - Paris 11)

In this article we consider the estimation of the joint distribution of the random coefficients and error term in the nonparametric random coefficients binary choice model. In this model from economics, each agent has to choose between two mutually exclusive alternatives based on the observation of attributes of the two alternatives and of the agents, the random coefficients account for unobserved heterogeneity of preferences. Because of the scale invariance of the model, we want to estimate the density of a random vector of Euclidean norm 1. If the regressors and coefficients are independent, the choice probability conditional on a vector of $d-1$ regressors is an integral of the joint density on half a hyper-sphere determined by the regressors. Estimation of the joint density is an ill-posed inverse problem where the operator that has to be inverted in the so-called hemispherical transform. We derive lower bounds on the minimax risk under $\xL^p$ losses and smoothness expressed in terms of Besov spaces on the sphere $\mathbb{S}^{d-1}$. We then consider a needlet thresholded estimator with data-driven thresholds and obtain adaptivity for $\xL^p$ losses and Besov ellipsoids under assumptions on the random design.

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Paper provided by HAL in its series Working Papers with number inria-00601274.

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Date of creation: Jun 2011
Date of revision:
Publication status: Published in [Research Report] RR-7647, INRIA. 2011
Handle: RePEc:hal:wpaper:inria-00601274
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  1. 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.
  2. Eric Gautier & Yuichi Kitamura, 2009. "Nonparametric Estimation in Random Coefficients Binary Choice Models," Cowles Foundation Discussion Papers 1721, Cowles Foundation for Research in Economics, Yale University.
  3. 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.
  4. Eric Gautier & Stefan Soderlein, 2011. "Estimating the Distribution of Treatment Effects," Working Papers 2011-25, Centre de Recherche en Economie et Statistique.
  5. repec:oup:restud:v:49:y:1982:i:3:p:403-09 is not listed on IDEAS
  6. 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.
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