Degrees of freedom for piecewise Lipschitz estimators

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A representation of the degrees of freedom akin to Stein’s lemma is given for a class of estimators of a mean value
parameter in Rn. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that even
though the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimators
with discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subset
selection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimate
of the degrees of freedom based on the lasso solution path, which in turn can be used for estimating the risk of lasso-OLS.
A similar estimate is proposed for best subset selection. The usefulness of the risk estimates for selecting the number of variables
is demonstrated via simulations with a particular focus on lasso-OLS.
Original languageEnglish
JournalAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
Issue number2
Pages (from-to)819-841
Number of pages23
Publication statusPublished - 1 May 2018


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