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On High-Dimensional Constrained Maximum Likelihood Inference.

Yunzhang Zhu1, Xiaotong Shen2, Wei Pan3

  • 1Department of Statistics, Ohio State University, Columbus, OH.

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|August 14, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a constrained maximum likelihood method to improve statistical inference in high-dimensional data. The approach effectively handles regularization impacts, offering better hypothesis testing for complex models and applications like brain network analysis.

Keywords:
(p, n)-asymptoticsBrain networksGeneralized Wilks phenomenonHigh-dimensionalityL0-regularizationSimilar tests

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Area of Science:

  • Statistics
  • Machine Learning
  • Computational Biology

Background:

  • High-dimensional data inference often faces challenges due to overparameterization and regularization.
  • Current methods like regularized models or debiasing have limitations, ignoring uncertainty or increasing variance.
  • Addressing regularization's impact on statistical inference is crucial for reliable high-dimensional analysis.

Purpose of the Study:

  • To propose a novel constrained maximum likelihood method for hypothesis testing in high-dimensional settings.
  • To alleviate the negative impacts of regularization on statistical inference, particularly for unspecific nuisance parameters.
  • To develop a method that allows parameter dimensions to increase with sample size, extending classical statistical phenomena.

Main Methods:

  • A constrained maximum likelihood approach is proposed, unregularizing hypothesized parameters while regularizing nuisance parameters via an L0-constraint.
  • The method is analogous to semiparametric likelihood inference in high-dimensional scenarios.
  • Conditions for establishing the asymptotic distribution of the constrained likelihood ratio are derived for Gaussian graphical models and linear regression.

Main Results:

  • The asymptotic distribution of the constrained likelihood ratio is established, permitting parameter dimensions to increase with sample size.
  • The limiting distribution is shown to be chi-square or normal, depending on the test's co-dimension, leading to asymptotically similar tests.
  • Numerical simulations demonstrate the proposed method's strong performance against competitors across various scenarios.

Conclusions:

  • The proposed constrained maximum likelihood method offers an effective way to handle regularization in high-dimensional inference.
  • The method provides a robust framework for hypothesis testing, extending the classical Wilks phenomenon.
  • The approach is successfully applied to brain network analysis, distinguishing Alzheimer's disease patients from healthy subjects using MRI data.