Robust parameter estimation and variable selection in regression models for asymmetric heteroscedastic data

In many real-world scenarios, not only the location but also the scale and even the skewness of the response variable may be influenced by explanatory variables. To achieve accurate predictions in such cases, it is essential to model location, scale, and skewness simultaneously. The joint location, scale, and skewness model of the skew-normal distribution is particularly useful for such data, as it relaxes the normality assumption, allowing for skewness. However, the estimation methods commonly used in these models tend to rely on classical approaches that are sensitive to outliers. Another challenge is selecting relevant variables. This study addresses these issues by first employing the maximum Lq-likelihood estimation method, which provides robust parameter estimation across the model. We then introduce the penalized Lq-likelihood method to select significant variables in the three sub-models. To obtain parameter estimates efficiently, we use the expectation-maximization algorithm. Through simulation studies and applications to real datasets, we demonstrate that the proposed methods outperform classical approaches, especially in the presence of outliers.