Normal Mode Copulas for Nonmonotonic Dependence

Copulas are helpful in studying joint distributions of two variables, in particular, when confounders are unobserved. However, most conventional copulas cannot model joint distributions where one variable does not increase or decrease in the other in a monotonic manner. For instance, suppose that two variables are linearly positively correlated for one type of unit and negatively for another type of unit. If the type is unobserved, we can observe only a mixture of both types. Seemingly, one variable tends to take either a high or low value (or a middle value) when the other variable is small (large), or vice versa. To address this issue, I consider an overlooked copula with trigonometric functions (Chesneau [2021, Applied Mathematics, 1(1), pp. 3–17]) that I name the “normal mode copula.” I apply the copula to a dataset about government formation and duration to demonstrate that the normal mode copula has better performance than other conventional copulas.