Nonparametric Maximum Likelihood Methods for Binary Response Models With Random Coefficients

Abstract

The venerable method of maximum likelihood has found numerous recent applications in nonparametric 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.

Keywords:

• Binary data analysis
• Convex optimization
• Empirical Bayes
• Hyperplane arrangements

Supplementary Materials

Two supplementary tables are provided in this section. Table 5 reports log-likelihood values for various choices of the tuning parameters of the Gautier–Kitamura estimator for the modal choice application. The contour plots for the Gautier–Kitamura estimates appearing in the main text are based on tuning parameters maximizing log-likelihood as reported in this table. Table 6 reports the location and mass of the NPMLE estimates for each subsample of the modal choice data; only points with mass greater than 0.001 are reported. Note, once again, that locations are arbitrary interior points within the polygons optimizing the log-likelihood.

Acknowledgments

The authors would like to express their appreciation to Frederico Ardila for his guidance toward some of the relevant combinatorial geometry literature, and to Steve Cosslett, Hide Ichimura, and Yuichi Kitamura for their pioneering work on the random coefficient binary response model. Thomas Stringham provided very capable research assistance. We also would like to thank Ismael Mourifie, Yuanyuan Wan, Miroslav Rada, Michal Černý, and Stanislav Volgushev for useful discussions.

Additional information

Funding

Jiaying Gu would like to acknowledge support from a SSHRC Insight Development grant.