The Influence Function of Graphical Lasso Estimators

TL;DR

This paper analyzes the robustness of graphical lasso estimators by deriving their influence functions and asymptotic properties, highlighting their sensitivity to outliers in multivariate analysis.

Contribution

It provides a theoretical comparison of robust and non-robust graphical lasso estimators through influence functions and asymptotic analysis.

Findings

Robust estimators have bounded influence functions.

Sensitivity curves indicate improved outlier resistance.

Asymptotic variances differ between estimators.

Abstract

The precision matrix that encodes conditional linear dependency relations among a set of variables forms an important object of interest in multivariate analysis. Sparse estimation procedures for precision matrices such as the graphical lasso (Glasso) gained popularity as they facilitate interpretability, thereby separating pairs of variables that are conditionally dependent from those that are independent (given all other variables). Glasso lacks, however, robustness to outliers. To overcome this problem, one typically applies a robust plug-in procedure where the Glasso is computed from a robust covariance estimate instead of the sample covariance, thereby providing protection against outliers. In this paper, we study such estimators theoretically, by deriving and comparing their influence function, sensitivity curves and asymptotic variances.