Questions concerning copulas

Question

I am new to the topic of copulas and my math is limited. I have different questions:

1. Is it correct to say, that $C: [0,1]^d \rightarrow [0,1]$ is a mapping from a multidimensional distribution to its one dimensional margins? Or that this is a mapping from the unit cube to the uniform margins?

2. I am often reading about the Fréchet-Hoeffding bounds, these seem to basically give the co-domain of the copula? Or what do they state and why do I need them? When are they reached? If I understood it correctly, than the lower bound is reached, when d=2 and the random variables are perfectly negatively dependent and the upper bound is reached, when the random variables are perfectly positively dependent?

3. In every book about copulas I read about the Sklar's Theorem. I do not understand the intuition of this mathematical theorem. What does it say and why do I need it in the context of copulas?

4. The definition of a copula is: It is a multivariate distribution with uniform marginal distributions, but I thought, that I can take any distribution for the marginal distributions and not only the uniform distribution?

I know these are beginners questions, but I could not find a good introductory book about copulas. Every book in this topic gives a lot of mathematical definitions and theorems and I am not on this level, unfortunately.

Answer 1

You might find Harry Joe's book easier going.

No, the mapping is not to the margins. It represents that the CDF is a function of its arguments - for every $(u_1,...,u_d)$ you get a single value for $P(U_1≤u_1,U_2≤u_2,...,U_d≤u_d)$; this is what $C$ is/does.

Q4 confuses copulas (which have uniform margins) with what you use them for (representing multivariate dependence more generally); you could transform the uniform margins of a copula to anything, but then the multivariate distribution is no longer itself a copula (but it still has the original copula)

Sklar's theorem is the thing that connects a distribution to its copula - in effect it explains what's going on in your Q4. Let's look at 2D. If distribution $H(x,y)$ has marginals $F(x)$ and $G(y)$ then Sklar says: $H(x,y)=C(F(x),G(y))$. Think about what $H$ is! What does the value of $H$ at some $(x,y)$ tell you?

Sklar's theorem is in effect telling you that if you take something with cdf $H$ and transform all its margins, you actually get $C$; the copula takes identical values to $H$ if you evaluate $C$ at $F(x)$ and $G(y)$.

Fréchet-Hoeffding bounds: Like any distribution function, $C$ is bounded between $0$ and $1$, but because of the additional restrictions on $C$, $C(\mathbf{0})=0$ and $C(\mathbf{1})=1$ (where $\mathbf{0} = (0,0,...,0)$ and similarly for $\mathbf{1}$). Fréchet-Hoeffding identifies the biggest and smallest values $C$ can have everywhere in between; the special restrictions on $C$ let us find bounds on $C$ for any $\mathbf{u}$. When $C$ takes its maximum value, yes, it corresponds to perfect monotonic dependence. In 2D, yes, the minimum value $C$ can take corresponds to perfect negative monotonic dependence.

Many of your confusions seem to stem from not really thinking about what $C$ or $H$ do. If you think about it in a very elementary sense, functions have a value; thinking about what it's the value of will help.

(Some more-or-less introductory papers and presentations that may help: 1 2 3 4 5 6 7 8)