The influence function of penalized regression estimators

Viktoria Öllerer*, Christophe Croux, Andreas Alfons

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

12 Citations (Scopus)

Abstract

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.

Original languageEnglish
Pages (from-to)741-765
Number of pages25
JournalStatistics
Volume49
Issue number4
DOIs
Publication statusPublished - 4 Jul 2015

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