Asymmetric linkages: maxmin vs. reflected maxmin copulas

In this paper we introduce some new copulas emerging from shock models. It was shown earlier that reflected maxmin copulas (RMM for short) are not just some specific singular copulas; they contain many important absolutely continuous copulas including the negative quadrant dependent part of the Eyraud-Farlie-Gumbel-Morgenstern class. The main goal of this paper is to develop the RMM copulas with dependent endogenous shocks and give evidence that RMM copulas may exhibit some characteristics better than the original maxmin copulas (MM for short): (1) An important evidence for that is the iteration procedure of the RMM transformation which we prove to be always convergent and we give many properties of it that are useful in applications. (2) Using this result we find also the limit of the iteration procedure of the MM transformation thus answering a question proposed earlier by Durante, Omladi\v{c}, Ora\v{z}em, and Ru\v{z}i\'{c}. (3) We give the multivariate dependent RMM copula that compares to the MM version given by Durante, Omladi\v{c}, Ora\v{z}em, and Ru\v{z}i\'{c}. In all our copulas the idiosyncratic and systemic shocks are combined via asymmetric linking functions as opposed to Marshall copulas where symmetric linking functions are used.