| By: |
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 (INRIA Saclay - Ile de France - SELECT - INRIA - Université Paris Sud - Paris XI - CNRS : UMR, Département de Mathématiques-Université de Paris X1 - Université Paris Sud - Paris XI) |
| Abstract: |
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. |
| Keywords: |
Discrete choice models;random coefficients; inverse problems; minimax rate optimality; adaptation; needlets; data-driven thresholding. |
| Date: |
2011–06 |
| URL: |
http://d.repec.org/n?u=RePEc:hal:wpaper:inria-00601274&r=dcm |