Modeling correlated ordinal data through Student's t copula

The t copula is a member of the family of elliptical copulas, which includes, among others, the Gaussian copula, with which it shares many properties. Unlike the latter, which is tail-independent, the t copula always exhibits equal and non-null lower and upper tail-dependence coefficients, even when its components are uncorrelated. Although the use of copulas is less straightforward in the discrete case, primarily due to theoretical challenges related to Sklar’s theorem, several contributions in the statistical literature have explored the use of the Gaussian copula for constructing multivariate discrete distributions. The use of the t copula is illustrated for modeling bivariate correlated discrete data by investigating two aspects. First, it is examined how its additional parameter, the degrees of freedom, can lead to different bivariate discrete distributions when the correlation parameter is held fixed. To quantify the differences between distributions, suitable distance measures are employed. Second, for a fixed (positive) correlation, a fixed number of degrees of freedom, and a fixed cardinality k of the support of the two discrete distributions, it is numerically identified which common k-point probability distribution induces the maximum linear correlation. It is found that the resulting correlation between the discrete random variables can exceed the copula's correlation parameter: A phenomenon that contrasts with the behavior observed when using the Gaussian copula.