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ESTIMATION OF FUNCTIONALS OF SPARSE COVARIANCE MATRICES.

Jianqing Fan1, Philippe Rigollet2, Weichen Wang1

  • 1Department of Operations Research and Financial Engineering Princeton University Princeton, New Jersey 08544 USA.

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Summary

Estimating functions of sparse correlation matrices is crucial for high-dimensional statistical tests. Simple thresholded estimators achieve optimal performance, adapting to sparsity and showing an elbow phenomenon in minimax rates.

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Area of Science:

  • Statistics
  • Econometrics
  • Machine Learning

Background:

  • High-dimensional statistical tests often simplify by ignoring correlations, leading to null distributions dependent on correlation matrix functionals like Frobenius and other ℓ norms.
  • Computing critical values for these tests necessitates understanding the estimation difficulty of such functionals, especially for sparse correlation matrices.

Purpose of the Study:

  • To investigate the estimation difficulty of functionals of sparse correlation matrices.
  • To develop and analyze plug-in procedures for estimating these functionals.
  • To determine if these procedures are sparsity-adaptive and minimax optimal.

Main Methods:

  • Developed simple plug-in procedures based on thresholded estimators of correlation matrices.
  • Analyzed the theoretical properties of these estimators, focusing on sparsity-adaptivity and minimax optimality.
  • Investigated the presence of the elbow phenomenon in minimax rates for functional estimation.

Main Results:

  • Demonstrated that thresholded estimators are sparsity-adaptive and minimax optimal for a broad class of sparse correlation matrices.
  • Confirmed the existence of an elbow phenomenon in the minimax rates, similar to previous functional estimation studies.
  • Validated the findings through simulations and an empirical study on financial econometrics data.

Conclusions:

  • Simple plug-in methods using thresholded estimators are effective for estimating functionals of sparse correlation matrices.
  • These methods offer optimal statistical performance, adapting to matrix sparsity.
  • The findings have implications for computing critical values in high-dimensional statistical tests and analyzing financial data.