Bootstrapping Lasso estimators

In this thesis we review the basic properties of the Bridge estimator of the regression parameter in a multiple linear regression model, which is the re- gression estimator that minimizes the residual sum of squares plus a penalty term of the form sum|β j |^γ . We focus on its asymptotic behaviour, first by studying consistency and then its asymptotic distribution. Bridge estima- tors include the particular case of Lasso estimator, which corresponds to the case γ = 1. This estimator has the attractiveness of being a model selection tool, therefore, it is particularly useful for identifying, in models with a high number of parameters, the most significant ones. In the second part of this thesis we review the main results about asymp- totic distribution of the bootstrapped Lasso estimator. Under some regular- ity conditions, the bootstrap approximation converges weakly to a random probability measure and so it fails to be consistent. We then report an alternative bootstrap method able to provide a valid approximation to the distribution of the Lasso estimator.