Abstract

Penalized least squares (or maximum likelihood) estimators, such as the famous LASSO , have been studied intensively in the last few years. While many properties of these estimators are now well understood, the understanding of their distributional characteristics, such as finite-sample and large-sample limit distributions, is still incomplete.

We study the distribution of the adaptive LASSO estimator (a variant of the LASSO introduced by Zou, 2006) for an orthogonal normal linear regression model in finite samples as well as in the large-sample limit. We show that these distributions are typically highly non-normal regardless of the choice of tuning of the estimator. The uniform convergence rate is obtained and shown to be slower than $n^{-1/2}$ in case the estimator is tuned to perform consistent model selection. Moreover, we derive confidence intervals based on the adaptive LASSO , compare their lenghts to the one based on the unpenalized estimator and also discuss the questionable statistical relevance of the ‘oracle’-property of this estimator. Finally, we provide an impossibility result regarding the estimation of the distribution function of the adaptive LASSO estimator.

joint work with B.M. Potscher (University of Vienna)

http://stochastik.math.uni-goettingen.de/~schneider/