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December 2018 A new scope of penalized empirical likelihood with high-dimensional estimating equations
Jinyuan Chang, Cheng Yong Tang, Tong Tong Wu
Ann. Statist. 46(6B): 3185-3216 (December 2018). DOI: 10.1214/17-AOS1655

Abstract

Statistical methods with empirical likelihood (EL) are appealing and effective especially in conjunction with estimating equations for flexibly and adaptively incorporating data information. It is known that EL approaches encounter difficulties when dealing with high-dimensional problems. To overcome the challenges, we begin our study with investigating high-dimensional EL from a new scope targeting at high-dimensional sparse model parameters. We show that the new scope provides an opportunity for relaxing the stringent requirement on the dimensionality of the model parameters. Motivated by the new scope, we then propose a new penalized EL by applying two penalty functions respectively regularizing the model parameters and the associated Lagrange multiplier in the optimizations of EL. By penalizing the Lagrange multiplier to encourage its sparsity, a drastic dimension reduction in the number of estimating equations can be achieved. Most attractively, such a reduction in dimensionality of estimating equations can be viewed as a selection among those high-dimensional estimating equations, resulting in a highly parsimonious and effective device for estimating high-dimensional sparse model parameters. Allowing both the dimensionalities of model parameters and estimating equations growing exponentially with the sample size, our theory demonstrates that our new penalized EL estimator is sparse and consistent with asymptotically normally distributed nonzero components. Numerical simulations and a real data analysis show that the proposed penalized EL works promisingly.

Citation

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Jinyuan Chang. Cheng Yong Tang. Tong Tong Wu. "A new scope of penalized empirical likelihood with high-dimensional estimating equations." Ann. Statist. 46 (6B) 3185 - 3216, December 2018. https://doi.org/10.1214/17-AOS1655

Information

Received: 1 March 2017; Revised: 1 October 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 1408.62053
MathSciNet: MR3852649
Digital Object Identifier: 10.1214/17-AOS1655

Subjects:
Primary: 62G99
Secondary: 62F40

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6B • December 2018
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