The Annals of Statistics

A new scope of penalized empirical likelihood with high-dimensional estimating equations

Jinyuan Chang, Cheng Yong Tang, and Tong Tong Wu

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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.

Article information

Source
Ann. Statist., Volume 46, Number 6B (2018), 3185-3216.

Dates
Received: March 2017
Revised: October 2017
First available in Project Euclid: 11 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1536631271

Digital Object Identifier
doi:10.1214/17-AOS1655

Mathematical Reviews number (MathSciNet)
MR3852649

Subjects
Primary: 62G99: None of the above, but in this section
Secondary: 62F40: Bootstrap, jackknife and other resampling methods

Keywords
Empirical likelihood estimating equations high-dimensional statistical methods moment selection penalized likelihood

Citation

Chang, Jinyuan; Tang, Cheng Yong; Wu, Tong Tong. A new scope of penalized empirical likelihood with high-dimensional estimating equations. Ann. Statist. 46 (2018), no. 6B, 3185--3216. doi:10.1214/17-AOS1655. https://projecteuclid.org/euclid.aos/1536631271


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Supplemental materials

  • Supplement to “A new scope of penalized empirical likelihood with high-dimensional estimating equations”. Additional technical proofs and a data analysis are given the Supplementary Material.