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Testability of high-dimensional linear models with nonsparse structures.

Jelena Bradic1, Jianqing Fan2, Yinchu Zhu3

  • 1Department of Mathematics, Halicioğlu Data Science Institute, University of California, San Diego.

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|July 11, 2022
PubMed
Summary
This summary is machine-generated.

Statistical inference in high-dimensional models is challenging. Problem difficulty hinges on covariate precision matrix sparsity, not regression coefficient sparsity, impacting testability and feature correlation tradeoffs.

Keywords:
Confidence intervalsMinimax theoryPrimary 62C20, 62F03Secondary 62F30, 62J07Uniform non-testabilityℓ2-constraint

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Area of Science:

  • Statistics
  • High-Dimensional Statistics
  • Statistical Inference

Background:

  • Recent interest in statistical inference for high-dimensional models.
  • Understanding limitations of statistical tests in complex models.

Purpose of the Study:

  • Investigate factors influencing the difficulty of statistical inference in high-dimensional settings.
  • Develop new concepts to analyze the limitations of hypothesis testing.
  • Establish new minimax testability results.

Main Methods:

  • Introduction of uniform and essentially uniform non-testability concepts.
  • Analysis of the precision matrix of covariates.
  • Development of minimax lower bounds and test constructions.

Main Results:

  • Inference difficulty depends on covariate precision matrix row sparsity, not regression coefficient sparsity.
  • New minimax testability results are independent of regression parameter sparsity.
  • Identified tradeoffs between testability and feature correlation.

Conclusions:

  • The sparsity of the precision matrix of covariates is a key determinant of statistical inference difficulty.
  • New theoretical frameworks provide sharper insights into hypothesis testing limitations.
  • Achievable minimax lower bounds are possible even with weak feature correlations.