Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with latent variables, and computing inference using the resulting augmented posterior. To evaluate this we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis-Hasting step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions we estimate 16 dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model.

The objective of this article is to provide the analyst with the necessary tools that allow for a robust ordering of joint distributions of health and income. We contribute to the literature on the measurement and inference of socioeconomic health inequality in three distinct but complementary ways. First, we provide a formalization of the socioeconomic health inequality-specific ethical principle introduced by Erreygers et al. (2012) . Second, we propose new graphical tools and dominance tests for the identification of robust orderings of joint distributions of income and health associated with this new ethical principle. Finally, based on both pro-poor and pro-extreme ranks ethical principles we address a very important aspect of dominance literature: the inference. To illustrate the empirical relevance of the proposed approach, we compare joint distributions of income and a health-related behavior in the United States in 1997 and 2014.

The health concentration curve is the standard graphical tool to depict socioeconomic health inequality in the literature on health inequality. This paper shows that testing for the absence of socioeconomic health inequality is equivalent to testing if the conditional expectation of health on income is a constant function that is equal to average health status. In consequence, any test for parametric specification of a regression function can be used to test for the absence of socioeconomic health inequality (subject to regularity conditions). Furthermore, this paper illustrates how to test for this equality using a test for parametric regression functional form and applies it to health-related behaviors from the National Health Survey 2014.

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