Skewed and heavy-tailed data can now pick an undirected graph: Student copula gives a zero-partial-correlation necessary condition

Undirected graphical models have relied essentially on Gaussian distributions, which fail when marginals are skewed or heavy-tailed; the pairwise Markov property can now be characterized through copula densities, and under a Student copula it yields a necessary condition of zero partial correlation. The Gaussian copula has null tail dependence coefficients, so it is unfitting when the probability of joint extremes stays high regardless of the extreme threshold; the multivariate t-distribution approach is not a Markov network in the strict sense, because its density does not factor according to the graph and a missing edge does not imply conditional independence. The derivation characterizes the pairwise Markov property in terms of copula densities and applies it to Gaussian, Student and Archimedean families; the Student case yields the necessary condition of zero partial correlation, while the Clayton copula shows that Archimedean families with few real parameters are incompatible with the pairwise Markov property.

Graph selection is carried out by penalized maximum likelihood, estimating the precision matrix and the degrees of freedom together so that the conditional independence structure can be recovered under non-Gaussian marginals. The Student t distribution is written as a scale mixture, a normal variable divided by an independent Gamma latent variable, so the missing latent variables can be handled by an ECM algorithm. Simulations draw two sets of 2000 observations from a 10-dimensional Student copula, one with Student marginals and one mixing normal, Student, lognormal, gamma, Weibull, chi-squared, beta and logistic marginals that are symmetric, skewed and heavy-tailed; the first set converges in 24 iterations and recovers the true graph perfectly, the second converges after relaxing the stopping rule and identifies most missing edges, with glasso as a comparison.

The result addresses undirected graphical model selection for continuous random vectors whose marginals are skewed or heavy-tailed and whose joint extreme probabilities do not fall with the threshold; under a Student copula it supplies the necessary condition of zero partial correlation together with a penalized maximum likelihood and ECM graph-selection procedure. Evaluation is limited to 10-dimensional simulations with two sets of 2000 observations, one with Student marginals and one with mixed skewed heavy-tailed marginals, compared against glasso. A reader following up could watch how the necessary condition behaves in higher dimensions or on real data.

Under the Student copula the pairwise Markov property yields only a necessary condition of zero partial correlation, not a sufficient one. The second, mixed-marginal dataset did not converge under the original stopping rule and converged only after relaxing it, and at a given discrepancy level some missing edges remained unidentified and one true edge was not recovered. Archimedean copulas with few real parameters are incompatible with the pairwise Markov property, so that family is outside this route.