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In this paper, we study the identifiability and the estimation of theparameters of a copula-based multivariate model when the margins are unknownand are arbitrary, meaning that they can be continuous, discrete, or mixturesof continuous and discrete. When at least one margin is not continuous, therange of values determining the copula is not the entire unit square and thissituation could lead to identifiability issues that are discussed here. Next,we propose estimation methods when the margins are unknown and arbitrary, usingpseudo log-likelihood adapted to the case of discontinuities. In view ofapplications to large data sets, we also propose a pairwise composite pseudolog-likelihood. These methodologies can also be easily modified to cover thecase of parametric margins. One of the main theoretical result is an extensionto arbitrary distributions of known convergence results of rank-basedstatistics when the margins are continuous. As a by-product, under smoothnessassumptions, we obtain that the asymptotic distribution of the estimationerrors of our estimators are Gaussian. Finally, numerical experiments arepresented to assess the finite sample performance of the estimators, and theusefulness of the proposed methodologies is illustrated with a copula-basedregression model for hydrological data. The proposed estimation is implementedin the R package CopulaInference, together with a function for checkingidentifiability.

Identifiability and inference for copula-based semiparametric models for random vectors with arbitrary marginal distributions