Copulas are a flexible way of modeling dependence in a set of variables where the association between variables can be elicited separately from the marginal distributions. A semi-parametric approach for estimating the dependence structure of bivariate distributions derived from copulas is investigated when the associated marginals are mixed, that is, consisting of both discrete and continuous components. The semi-parametric likelihood approach is proposed for obtaining the estimator of the dependence parameter under unknown marginals. The consistency and asymptotic normality of the estimator is established as sample size tends to infinity. For constructing confidence intervals in practice, an estimator of the asymptotic variance is proposed and its properties are investigated via simulation. Extensions to higher dimensions are discussed. Several simulation studies and real data examples are provided for investigating the application of the developed methodology of inference. This work generalizes prior results obtained on the estimation of dependence when the marginals are continuous by Genest et al (1995).
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