This thesis deals examines the adaptive LASSO estimator in the setting of moving parameter in the low-dimensional case, while the tuning parameters may vary over the components. The main part deals with the construction of asymptotic confidence sets based on the adaptive LASSO estimator in the case where at least one component of the tuning parameter is tuned to perform consistent model selection. The asymptotic distribution of the appropriately scaled and centered adaptive LASSO estimator is derived implicitly as the minimizer of a stochastic function, which is used to create confidence sets with asymptotically infimal coverage probability of 1. Besides confidence sets of the partially consistent tuned adaptive LASSO estimator, a condition on the tuning parameters is shown to be equivalent to consistency in parameter estimation. Conditions concerning the consistency in model selection are also derived. In particular, obtaining consistency in model selection for the adaptive LASSO estimator requires consistency in parameter estimation.