The adaptive Lasso in high dimensional sparse heteroscedastic models
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Date
2011-07-04
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Abstract
In this paper we study the asymptotic properties of the adaptive Lasso estimate in high dimensional sparse linear regression models with heteroscedastic errors. It is demonstrated that model selection properties and asymptotic normality of the selected parameters remain valid but with a suboptimal asymptotic variance. A weighted adaptive Lasso estimate is introduced and is investigated. In particular, it is shown that the new estimate performs consistent model selection and that linear combinations of the estimates corresponding to the
non-vanishing components are asymptotically normally distributed with a smaller variance
than those obtained by the "classical" adaptive Lasso. The results are illustrated in a data example and by means of a small simulation study.
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heteroscedasticity, high dimensional data, penalized regression, variable selection