I am currently working with 3 dimensions, and have some functions $C(u_1, u_2, u_3)$ for which I need to check whether or not they are a copula.

What are all the requirements that I need to check? In my book, it says I need to check some sort of rectangle inequality, but it is presented in a very general and unclear format. I would like a simpler representation for the particular case of 3 dimensions.

Or is there perhaps an easier way than using this rectangle inequality?

  • $\begingroup$ You can interpret the rectangle inequality as the "non-decreasing" property of a CDF, which means you have to show that your $C$ is non-decreasing in every dimension. as $u_i$ increases, with $u_j$ and $u_k$ being constant, $C(u_i,u_j,u_k)$ must be non-decreasing for all $u_i$, $u_j$ and $u_k$. $\endgroup$ – Kiran K. Dec 28 '18 at 17:27
  • $\begingroup$ Proving that 3-dimensional copula is a copula can be tedious in some cases, since you need to make sure that the "non-decreasing" property (or in other words, "positive volume of the rectangle") holds for any choice of ($u_1, u_2, u_3$). Disproving can be somewhat easier, since you only need to find a counterexample case. $\endgroup$ – Nutle Jan 2 at 13:20

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