Right-truncated Archimedean and related copulas

Abstract

The copulas of random vectors with standard uniform univariate margins truncated from the right are considered and a general formula for such right-truncated conditional copulas is derived. This formula is analytical for copulas that can be inverted analytically as functions of each single argument. This is the case for Archimedean and related copulas. The resulting right-truncated Archimedean copulas are not only analytically tractable but can also be characterized as tilted Archimedean copulas, one of the main contributions of this work. This characterization now allows one to not only immediately obtain a limiting Clayton copula for a general vector of truncation points converging to zero (an extension of a result known in the special case of equal truncation points), but to work with an exact model instead of an approximate limiting model in the first place. As tilted Archimedean copulas have been studied in the literature, the characterization allows one to obtain various analytical and stochastic properties of right-truncated Archimedean copulas. Further contributions include the characterization of right-truncated Archimax copulas with logistic stable tail dependence functions as tilted outer power Archimedean copulas, and an analytical form of right-truncated nested Archimedean copulas.