Flexible Modeling of Multivariate Skewed and Heavy-Tailed Data via a Non-Central Skew t Distribution: Application to Tumor Shape Data

Abstract: We propose a flexible formulation of the multivariate non-central skew t (NCST) distribution, defined by scaling skew-normal random vectors with independent chi-squared variables. This construction extends the classical multivariate t family by allowing both asymmetry and non-centrality, which provides an alternative to existing skew t models that often rely on restrictive assumptions for tractability. We derive key theoretical properties of the NCST distribution, which includes its moment structure, affine transformation behavior, and the distribution of quadratic forms. Due to the lack of a closed-form density, we implement a Monte Carlo likelihood approximation to enable maximum likelihood estimation and evaluate its performance through simulation studies. To demonstrate practical utility, we apply the NCST model to breast cancer diagnostic data, modeling multiple features of tumor shape. The NCST model achieves a superior fit based on information criteria and visual diagnostics, particularly in the presence of skewness and heavy tails compared to standard alternatives, including the multivariate normal, skew normal, and Azzalini's skew $t$ distribution. Our findings suggest that the NCST distribution offers a useful and interpretable choice for modeling complex multivariate data, which highlights promising directions for future development in likelihood inference, Bayesian computation, and applications involving asymmetry and non-Gaussian dependence.