Discriminating random variables on time-series on both their distribution and dependence information is motivated by the study of financial assets returns. For example, given two assets where their returns are perfectly correlated, are these returns always similar from a risk perspective? According to Kelly and Jiang (2014), the answer is no, because we did not take into account the distribution differences and distribution information. Therefore, there is a need for a distance metric that can distinguish underlying distributions of time-series even if they are perfectly correlated.