Degrees of freedom for piecewise Lipschitz estimators
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Degrees of freedom for piecewise Lipschitz estimators. / Mikkelsen, Frederik Riis; Hansen, Niels Richard.
In: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 54, No. 2, 01.05.2018, p. 819-841.Research output: Contribution to journal › Journal article › peer-review
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TY - JOUR
T1 - Degrees of freedom for piecewise Lipschitz estimators
AU - Mikkelsen, Frederik Riis
AU - Hansen, Niels Richard
PY - 2018/5/1
Y1 - 2018/5/1
N2 - A representation of the degrees of freedom akin to Stein’s lemma is given for a class of estimators of a mean valueparameter in Rn. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that eventhough the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimatorswith discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subsetselection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimateof 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 variablesis demonstrated via simulations with a particular focus on lasso-OLS.
AB - A representation of the degrees of freedom akin to Stein’s lemma is given for a class of estimators of a mean valueparameter in Rn. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that eventhough the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimatorswith discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subsetselection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimateof 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 variablesis demonstrated via simulations with a particular focus on lasso-OLS.
U2 - 10.1214/17-AIHP822
DO - 10.1214/17-AIHP822
M3 - Journal article
VL - 54
SP - 819
EP - 841
JO - Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
JF - Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
SN - 0246-0203
IS - 2
ER -
ID: 197001906