Generalized Precision Matrices for Non-gaussian Distributions: Theory and Portfolio Applications

We introduce a general measure of conditional local dependence for multivariate vectors and use it to define a generalized precision matrix (GPM) that is valid for any statistical distribution. We show that, in the Gaussian case, the GPM coincides with the inverse of the covariance matrix. Additionally, we derive the GPM analytically for the multivariate t-Student, multivariate skew-normal, and multivariate skew-t distributions. Using simulation, we compare the performance of the different estimators, discussing their properties. As a real-world application, we test the GPM within the Markowitz minimum variance portfolio framework, demonstrating that the multivariate skew-t model provides a superior fit during financial crisis periods