Identifiability and the Simplifying Assumption

A parameter in a statistical model is identifiable if it has a consistent estimator. We consider a conditional copula as a parameter in a regular vine model. As a baseline, no assumptions are made regarding bivariate margins. A non-simplified representation of a conditional copula expresses this copula as a convex combination of other copulas indexed by a measurable function of the conditioning variables. From this baseline three possibilities arise: (a) a non-simplified representation does not exist, (b) a non-simplified representation exists and is identifiable and (c) non-simplified representations exist but are not identifiable. The data generation process may either (1) sample the same conditional copula for each value of the conditioning variables, or (2) sample different conditional copulas for different values of the conditioning variables. The combinations (a.1), (b.1), (b.2), (c.1) and (c.2) are all possible. Moreover, different generation processes may produce the same joint distribution. In cases of (c) we may have strong evidence against the simplifying assumption, but need additional assumptions to render the non-simplified copula identifiable and define a consistent estimator. This article is an introductory exploration of these issues. Distinct continuous convex decompositions of the independent copula are found with different correlation functions. We find new extreme copulas on the unit square using bijective and non-bijective measure preserving maps, and conjecture that these constitute the set of extreme copula.