Nonparametric Maximum Likelihood Methods for Binary Response Models With Random Coefficients

The venerable method of maximum likelihood has found numerous recent applications in <i>nonparametric</i> estimation of regression and shape constrained densities. For mixture models the nonparametric maximum likelihood estimator (NPMLE) of Kiefer and Wolfowitz plays a central role in recent developments of empirical Bayes methods. The NPMLE has also been proposed by Cosslett as an estimation method for single index linear models for binary response with random coefficients. However, computational difficulties have hindered its application. Combining recent developments in computational geometry and convex optimization, we develop a new approach to computation for such models that dramatically increases their computational tractability. Consistency of the method is established for an expanded profile likelihood formulation. The methods are evaluated in simulation experiments, compared to the deconvolution methods of Gautier and Kitamura and illustrated in an application to modal choice for journey-to-work data in the Washington DC area. Supplementary materials for this article are available online.