Journal
ANNALS OF STATISTICS
Volume 46, Issue 6B, Pages 3185-3216Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/17-AOS1655
Keywords
Empirical likelihood; estimating equations; high-dimensional statistical methods; moment selection; penalized likelihood
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Funding
- Fundamental Research Funds for the Central Universities [JBK171121, JBK170161, JBK150501]
- NSFC [11501462]
- Center of Statistical Research at SWUFE
- Joint Lab of Data Science and Business Intelligence at SWUFE
- NSF [IIS-1546087, SES-1533956]
- NIH [R01HL094572, R01HL119058]
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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.
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