A Regularization Based Approach to Estimation of a Two Component Nonparametric Density Mixture with a Known Component

• Activity Number: 641 - Recent Advances in Density Mixture Modeling and EM-Like Algorithms: Frequentist and Bayesian Views
• Type: Topic Contributed
• Date/Time: Thursday, August 1, 2019 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Michael Levine* and Zuofeng Shang and Zhou Shen
• Companies: Purdue University and IUPUI and J.P. Morgan
• Keywords: penalized smoothed likelihood; MM algorithm; regularization

Abstract

We consider a semiparametric mixture of two density functions where only one of them is known. Such mixtures have a history of application to the problem of detecting differentially expressed genes under two or more conditions in microarray data. We do not assume any additional structure on the unknown density function (e.g. symmetry). For this mixture model, we derive a new sufficient identifiability condition. We also suggest a novel approach to estimation of this model that is based on an idea of applying a maximum smoothed likelihood to what would otherwise have been an ill-posed problem. We introduce an iterative MM (Majorization-Minimization) algorithm that estimates all of the model parameters. We establish that the algorithm possesses a descent property with respect to a log-likelihood type objective functional and prove that the algorithm, indeed, converges. We also show how to establish the large-sample convergence of the solution that the MM algorithm arrives at. Finally, we also illustrate the performance of our algorithm in a simulation study and using a real dataset. This is a joint work with Zuofeng Shang (IUPUI) and Zhou Shen (J.P. Morgan)