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October 2019 An operator theoretic approach to nonparametric mixture models
Robert A. Vandermeulen, Clayton D. Scott
Ann. Statist. 47(5): 2704-2733 (October 2019). DOI: 10.1214/18-AOS1762

Abstract

When estimating finite mixture models, it is common to make assumptions on the mixture components, such as parametric assumptions. In this work, we make no distributional assumptions on the mixture components and instead assume that observations from the mixture model are grouped, such that observations in the same group are known to be drawn from the same mixture component. We precisely characterize the number of observations $n$ per group needed for the mixture model to be identifiable, as a function of the number $m$ of mixture components. In addition to our assumption-free analysis, we also study the settings where the mixture components are either linearly independent or jointly irreducible. Furthermore, our analysis considers two kinds of identifiability, where the mixture model is the simplest one explaining the data, and where it is the only one. As an application of these results, we precisely characterize identifiability of multinomial mixture models. Our analysis relies on an operator-theoretic framework that associates mixture models in the grouped-sample setting with certain infinite-dimensional tensors. Based on this framework, we introduce a general spectral algorithm for recovering the mixture components.

Citation

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Robert A. Vandermeulen. Clayton D. Scott. "An operator theoretic approach to nonparametric mixture models." Ann. Statist. 47 (5) 2704 - 2733, October 2019. https://doi.org/10.1214/18-AOS1762

Information

Received: 1 October 2016; Revised: 1 March 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114926
MathSciNet: MR3988770
Digital Object Identifier: 10.1214/18-AOS1762

Subjects:
Primary: 62E10
Secondary: 62G05

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • October 2019
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