Adaptive lasso selection is a modification of lasso selection; in adaptive lasso selection, weights are applied to each of
the parameters in forming the lasso constraint (Zou, 2006). More precisely, suppose that the response has mean 0 and the regressors
are scaled to have mean 0 and common standard deviation. Furthermore, suppose that you can find a suitable estimator
of the parameters in the true model and you define a weight vector by
, where
. Then the adaptive lasso regression coefficients
are the solution to the following constrained optimization problem:
PROC HPREG uses the solution to the unconstrained least squares problem as the estimator . This is appropriate unless collinearity is a concern. If the regressors are collinear or nearly collinear, then Zou (2006)
suggests using a ridge regression estimate to form the adaptive weights.