Copulas Based on Marshall–Olkin Machinery

Abstract

We present a general construction principle for copulas that is inspired by the celebrated Marshall-Olkin exponential model. From this general construction method we derive special sub-classes of copulas that could be useful in different situations and recall their main properties. Moreover, we discuss possible estimation strategy for the proposed copulas. The presented results are expected to be useful in the construction of stochastic models for lifetimes (e.g. in reliability theory) or in credit risk models.

Constructing copulas from shock models with imprecise distributions

“…We have thus shown that both F u (x 1 +) F (x 1 +) and F u (x 1 −) F (x 1 −), whence by applying the argument of Case 1 of this proof, we derive Inequality (12). Together with (11) this implies the desired conclusion φ(u) φ (u) and completes the proof.…”

“…A substantial theory on shock model copulas has developed in the meantime including [5,6,15,23,34,45]. The third milestone on the path of shock model induced copulas was made in 2015 by F. Durante, S. Girard and G. Mazo [12,13] who describe a general construction principle for copulas based on shock models. In [13] a new possibility in the research of shock models induced copulas was introduced, i.e.…”

Copulae: An overview and recent developments

“…When the univariate distributions are all continuous, and there is only one shock that affects all the variables at once, then the construction of the Marshall–Olkin copula is quite simple using a function depending on the shock distribution, as shown in Durante et al (2016). Durante et al (2015) provide an idea of how to combine different Marshall–Olkin copulae. AC can also be applied to the study of Marshall–Olkin distributions because the resulting dependence structure can be modeled using the generator of a trivariate Archimedean copula.…”