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Abstract

This paper analyzes estimation by bootstrap variable-selection ina simple Gaussian model where the dimension of the unknown parameter mayexceed that of the data. A naive use of the bootstrap in this problemproduces risk estimators for candidate variable-selections that have astrong upward bias. Resampling from a less overfitted model removes the bias and leads to bootstrap variable-selections that minimize risk asymptotically. A related bootstrap technique generates confidence sets that are centered atthe best bootstrap variable-selection and have two further properties: theasymptotic coverage probability for the unknown parameter is as desired; andthe confidence set is geometrically smaller than a classical competitor.The results suggest a possible approach to confidence sets in other inverseproblems where a regularization technique is used.