Abstract
Variable selection is an important property of shrinkage methods. The adaptive lasso is an oracle procedure and can do consistent variable selection. In this paper, we provide an explanation that how use of adaptive weights make it possible for the adaptive lasso to satisfy the necessary and almost sufcient condition for consistent variable selection. We suggest a novel algorithm and give an important result that for the adaptive lasso if predictors are normalised after the introduction of adaptive weights, it makes the adaptive lasso performance identical to the lasso.
Highlights
Tibshirani(1996) proposed a new shrinkage method named least absolute shrinkage and selection operator, abbreviated as lasso
We show that in the situations when the necessary condition for the consistent variable selection fails for the lasso and if for the adaptive lasso predictors are normalised after the introduction of adaptive weights, the adaptive lasso performs identical to the lasso
Many other methods have been suggested in the literature but lasso-type methods are currently popular among researchers (Knight and Fu, 2000; Fan and Li, 2001; Wang and Leng, 2007; Hsu et al, 2008)
Summary
Tibshirani(1996) proposed a new shrinkage method named least absolute shrinkage and selection operator, abbreviated as lasso. The lasso can do consistent model selection if it satisfies a necessary condition on the covariance matrix of predictors (Zhao and Yu, 2006). This same condition is independently derived by Zou (2006). The oracle properties of these procedures are studied for different models and under various conditions e.g. the necessary condition for consistent selection discussed in Zhao and Yu (2006) and Zou (2006).
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