Estimation of large covariance and precision matrices from temporally dependent observations

Hai Shu, Bin Nan

Research output: Contribution to journalArticlepeer-review


We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that several commonly used methods for independent observations can be applied to the temporally dependent data. In particular, the rates of convergence are obtained for the generalized thresholding estimation of covariance and correlation matrices, and for the constrained l 1 minimization and the l 1 penalized likelihood estimation of precision matrix. Properties of sparsistency and sign-consistency are also established. A gap-block cross-validation method is proposed for the tuning parameter selection, which performs well in simulations. As a motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.

Original languageEnglish (US)
Pages (from-to)1321-1350
Number of pages30
JournalAnnals of Statistics
Issue number3
StatePublished - Jan 2019


  • Brain functional connectivity
  • Correlation matrix
  • Heavy tail
  • High-dimensional data
  • Long memory
  • Minimax optimal convergence rates
  • Nonstationarity
  • Sub-Gaussian tail
  • Temporal dependence

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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