  Modern variable selection procedures make use of penalization methods to
execute simultaneous model selection and estimation. A popular method is the
LASSO (least absolute shrinkage and selection operator), the use of which
requires selecting the value of a tuning parameter. This parameter is typically
tuned by minimizing the cross-validation error or Bayesian information
criterion (BIC) but this can be computationally intensive as it involves
fitting an array of different models and selecting the best one. In contrast
with this standard approach, we have developed a procedure based on the
so-called "smooth IC" (SIC) in which the tuning parameter is automatically
selected in one step. We also extend this model selection procedure to the
distributional regression framework, which is more flexible than classical
regression modelling. Distributional regression, also known as multiparameter
regression (MPR), introduces flexibility by taking account of the effect of
covariates through multiple distributional parameters simultaneously, e.g.,
mean and variance. These models are useful in the context of normal linear
regression when the process under study exhibits heteroscedastic behaviour.
Reformulating the distributional regression estimation problem in terms of
penalized likelihood enables us to take advantage of the close relationship
between model selection criteria and penalization. Utilizing the SIC is
computationally advantageous, as it obviates the issue of having to choose
multiple tuning parameters.
