Variable Selection Using a Smooth Information Criterion for Multi-Parameter Regression Models

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), which contains 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. However, 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 so-called "multi-parameter regression" framework, which is more flexible than classical regression modelling. Multi-parameter regression 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 multi-parameter 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.