User-Friendly Covariance Estimation for Heavy-Tailed Distributions

• Yuan Ke ^[1] ; Stanislav Minsker ^[2] ; Zhao Ren ^[3] ; Qiang Sun ^[4] ; Zhou, Wen-Xin ^[5]
  1. [1] University of Georgia

     University of Georgia

     Estados Unidos

     
  2. [2] University of Southern California

     University of Southern California

     Estados Unidos

     
  3. [3] University of Pittsburgh

     University of Pittsburgh

     City of Pittsburgh, Estados Unidos

     
  4. [4] University of Toronto

     University of Toronto

     Canadá

     
  5. [5] University of California

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• Localización: Statistical science, ISSN 0883-4237, Vol. 34, Nº. 3, 2019, págs. 454-471
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • We provide a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation.

    Specifically, we introduce elementwise and spectrumwise truncation operators, as well as their M-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness that is characterized by the breakdown property, we focus on the tail robustness which is evidenced by the connection between nonasymptotic deviation and confidence level. The key insight is that estimators should adapt to the sample size, dimensionality and noise level to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate practical implementation, we propose data-driven procedures that automatically calibrate the tuning parameters.

    We demonstrate their applications to a series of structured models in high dimensions, including the bandable and low-rank covariance matrices and sparse precision matrices. Numerical studies lend strong support to the proposed methods.