The univariate distorted distributions are a good tool to model risks, order statistics, coherent system lifetimes, etc. This concept is extended to model joint distributions of random vectors. The representations are similar to copula representations by changing the marginal distribution functions with arbitrary univariate distribution functions. The main advantage is that this representation has similar properties to that of copulas, it is more flexible and that, in some models, it allows to get simple representations. These facts are illustrated with several examples which include paired data, multivariate residual lifetimes, record values, order statistics and coherent systems. In particular, the representation can be used to predict these values by using quantile regression techniques. This work is partially supported by Ministerio de Ciencia e Innovación of Spain under grant PID2019-103971GB-I00/AEI/10.13039/501100011033.
