Title: On the Distribution, Model Selection Properties and Uniqueness of the Lasso Estimator in Low and High Dimensions,
Abstract: We derive expressions for the finite-sample distribution of the Lasso
estimator in the context of a linear regression model with normally distributed
errors in low as well as in high dimensions by exploiting the structure of the
optimization problem defining the estimator. In low dimensions we assume full
rank of the regressor matrix and present expressions for the cumulative
distribution function as well as the densities of the absolutely continuous
parts of the estimator. Additionally, we establish an explicit formula for the
correspondence between the Lasso and the least-squares estimator. We derive
analogous results for the distribution in less explicit form in high dimensions
where we make no assumptions on the regressor matrix at all. In this setting,
we also investigate the model selection properties of the Lasso and show that
possibly only a subset of models might be selected by the estimator, completely
independently of the observed response vector. Finally, we present a condition
for uniqueness of the estimator that is necessary as well as sufficient.