Extended Oracle Properties of Adaptive Lasso Estimators

Abstract

We study the asymptotic properties of adaptive lasso estimators when some components of the parameter of interest <i>&beta;</i> are strictly different than zero, while other components may be zero or may converge to zero with rate n^{- &delta;}, with &delta; >0 , where n denotes the sample size. To achieve this objective, we analyze the convergence/divergence rates of each term in the first-order conditions of adaptive lasso estimators. First, we derive conditions that allow selecting tuning parameters in order to ensure that adaptive lasso estimates of n^{- &delta;}-components indeed collapse to zero. Second, in this case, we also derive asymptotic distributions of adaptive lasso estimators for nonzero components. When &delta; >1/2 , we obtain the usual n^1/2-asymptotic normal distribution, while when 0< &delta; &le; 1/2, we show n^&delta;-consistency combined with (biased) n^{1/2- &delta;}-asymptotic normality for nonzero components. We call these properties, Extended Oracle Properties . These results allow practitioners to exclude in their model the asymptotically negligible variables and make inferences on the asymptotically relevant variables.