Simple expressions of the LASSO and SLOPE estimators in low-dimension

Patrick J C Tardivel (1) , Rémi Servien (1) , Didier Concordet (1)

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We study the LASSO and SLOPE estimators when the design X satisfies ker(X)=0. Similarly to the LASSO, the SLOPE estimator has an explicit expression when the design matrix X is orthogonal which is reported in the main theorem of this article. We state that, even if the design is not orthogonal, even if residuals are correlated, up to a transformation, the LASSO and SLOPE estimators have a simple expression based on the Best Linear Unbiased Estimator (BLUE). Comparisons with the LASSO estimator show the benefits of the soft-thresholded BLUE.

Mots clés: LASSO; Best linear unbiased estimator; SLOPE

Domaines: Statistiques [stat]; Mathématiques [math]

Patrick J C Tardivel, Rémi Servien, Didier Concordet. Simple expressions of the LASSO and SLOPE estimators in low-dimension. Statistics, 2020, ⟨10.1080/02331888.2020.1720019⟩. ⟨hal-01755076v3⟩