TY - JOUR
T1 - A conditional gradient approach for nonparametric estimation of mixing distributions
AU - Jagabathula, Srikanth
AU - Subramanian, Lakshminarayanan
AU - Venkataraman, Ashwin
N1 - Funding Information:
History: Accepted by Serguei Netessine, operations management. Funding: S. Jagabathula’s and A. Venkataraman’s research was supported in part by the National Science Foundation [Grant CMMI-1454310]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2019.3373.
Funding Information:
S. Jagabathula's and A. Venkataraman's research was supported in part by the National Science Foundation [Grant CMMI-1454310]. The authors thank the department editor, the associate editor, and the anonymous referees whose comments and feedback helped improve the manuscript greatly; and the conference participants at Manufacturing & Service Operations Management and INFORMS, the participants at the University of Minnesota Institute for Mathematics and Its Applications Data-Driven Supply Chain Management workshop; and the seminar participants at New York University Stern School of Business, University of Texas Dallas Naveen Jindal School of Management, and Northwestern Kellogg School of Management for their constructive comments that have helped improve the paper.
PY - 2020/8
Y1 - 2020/8
N2 - 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 a priori 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 (also known as 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 the 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 (16 times faster), in-sample fit (up to 24% reduction in the log-likelihood loss), and predictive (average 28% 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.
AB - 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 a priori 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 (also known as 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 the 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 (16 times faster), in-sample fit (up to 24% reduction in the log-likelihood loss), and predictive (average 28% 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.
KW - Consideration sets
KW - Convex optimization
KW - Mixture of logit
KW - Nonparametric estimation
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U2 - 10.1287/mnsc.2019.3373
DO - 10.1287/mnsc.2019.3373
M3 - Article
AN - SCOPUS:85090216762
SN - 0025-1909
VL - 66
SP - 3635
EP - 3656
JO - Management Science
JF - Management Science
IS - 8
ER -