A Conditional Gradient Approach for Nonparametric Estimation of Mixing Distributions

61 Pages Posted: 1 Sep 2017 Last revised: 21 Oct 2018

See all articles by Srikanth Jagabathula

Srikanth Jagabathula

New York University (NYU) - Department of Information, Operations, and Management Sciences

Lakshminarayanan Subramanian

New York University (NYU)

Ashwin Venkataraman

Harvard University - Institute for Quantitative Social Science; New York University (NYU) - Courant Institute of Mathematical Sciences, Students

Date Written: October 20, 2018

Abstract

Mixture models are versatile tools that are used extensively in many fields, including operations, marketing, and econometrics. The main challenge in estimating mixture models is that the mixing distribution is often unknown and imposing apriori parametric assumptions can lead to model misspecification issues. In this paper, we propose a new methodology for nonparametric estimation of the mixing distribution of a mixture of logit models. We formulate the likelihood-based estimation problem as a constrained convex program and apply the conditional gradient (a.k.a. Frank-Wolfe) algorithm to solve this convex program. We show that our method iteratively generates the support of the mixing distribution and the mixing proportions. Theoretically, we establish sublinear convergence rate of our estimator and characterize the structure of the recovered mixing distribution. Empirically, we test our approach on real-world datasets. We show that it outperforms the standard expectation-maximization (EM) benchmark on speed (16x faster), in-sample fit (up to 24% reduction in the log-likelihood loss), and predictive (average 27% reduction in standard error metrics) and decision accuracies (extracts around 23% more revenue). On synthetic data, we show that our estimator is robust to different ground-truth mixing distributions and can also account for endogeneity.

Keywords: nonparametric estimation, mixtures, conditional gradient, consideration sets

Suggested Citation

Jagabathula, Srikanth and Subramanian, Lakshminarayanan and Venkataraman, Ashwin, A Conditional Gradient Approach for Nonparametric Estimation of Mixing Distributions (October 20, 2018). Available at SSRN: https://ssrn.com/abstract=3029881 or http://dx.doi.org/10.2139/ssrn.3029881

Srikanth Jagabathula

New York University (NYU) - Department of Information, Operations, and Management Sciences ( email )

44 West Fourth Street
New York, NY 10012
United States

Lakshminarayanan Subramanian

New York University (NYU) ( email )

Bobst Library, E-resource Acquisitions
20 Cooper Square 3rd Floor
New York, NY 10003-711
United States

Ashwin Venkataraman (Contact Author)

Harvard University - Institute for Quantitative Social Science ( email )

1737 Cambridge Street
Cambridge, MA 02138
United States

New York University (NYU) - Courant Institute of Mathematical Sciences, Students ( email )

New York, NY
United States

Register to save articles to
your library

Register

Paper statistics

Downloads
125
rank
212,305
Abstract Views
638
PlumX