Likelihood Based Inference and Bias Reduction in the Modified Skew-t-Normal Distribution

In this paper, likelihood-based inference and bias correction based on Firth's approach are developed in the modified skew-t-normal (MStN) distribution. The latter model exhibits a greater flexibility than the modified skew-normal (MSN) distribution since it is able to model heavily skewed data and thick tails. In addition, the tails are controlled by the shape parameter and the degrees of freedom. We provide the density of this new distribution and present some of its more important properties including a general expression for the moments. The Fisher's information matrix together with the observed matrix associated with the log-likelihood are also given. Furthermore, the non-singularity of the Fisher's information matrix for the MStN model is demonstrated when the shape parameter is zero. As the MStN model presents an inferential problem in the shape parameter, Firth's method for bias reduction was applied for the scalar case and for the location and scale case.

Automated summary

In this paper, likelihood-based inference and bias correction using Firth's approach are developed for the modified skew-t-normal (MStN) distribution. Compared to the modified skew-normal (MSN) distribution, the MStN model allows for modeling heavily skewed data and thick tails. The shape parameter and the degrees of freedom control the tails. The paper provides the density and important properties of this new distribution and presents the Fisher's information matrix and observed matrix associated with the log-likelihood. Additionally, the non-singularity of the Fisher's information matrix for the MStN model is demonstrated when the shape parameter is zero. Firth's method for bias reduction is applied to the MStN model to address the inferential problem in the shape parameter, both in scalar and location-scale cases.