Robust and Sparse Estimation of the Inverse Covariance Matrix Using Rank Correlation Measures

Abstract

Spearman’s rank correlation is a robust alternative for the standard correlation coefficient. Using ranks instead of the actual values of the observations, the impact of outliers remains limited. In this paper, we study an estimator based on this rank correlation measure for estimating covariance matrices and their inverses. The resulting estimator is robust and consistent at the normal distribution. By applying the graphical lasso, the inverse covariance matrix estimator is positive definite if more variables than observations are available in the data set. Moreover, it will contain many zeros, and is therefore said to be sparse. Instead of Spearman’s rank correlation, one can use Kendall correlation, Quadrant correlation or Gaussian rank scores. A simulation study compares the different estimators. This type of estimator is particularly useful for estimating (inverse) covariance matrices in high dimensions, when the data may contain several outliers in many cells of the data matrix. More traditional robust estimators are not well defined or computable in this setting. An important feature of the proposed estimators is their simplicity and easiness to compute using existing software.