Decomposing a random variable into marginals and copula

Question

I’m having trouble getting understanding how to actual construct a copula, from my understanding it captures the purely joint features of a joint distribution. I’ve been working with the following example.

Let X,Y be random variables with joint distribution function $$H(x,y) = (1 + e^{-x} + e^{-y})^{-1}$$ I have got by letting x and y tend to infinity respectively that the marginals of X and Y are standard logistic distributions given by $$F(x) = (1 + e^{-x})^{-1}, G(y) = (1 + e^{-y})^{-1}$$ I have to show that the copula of X and Y is $$C(u,v) = \frac{uv}{u + v - uv}$$ but I don’t know how to go about doing this? I’ve tried working with Sklars theorem but I can’t seem to get my head around this.

Aside from this specific example what is the approach in general to retrieve copulas from joint distributions? Any help would be great, thanks!

Answer 1

I will take you through a set of simple steps that will work for continuous distributions. (A little extra care is needed to handle the jumps that occur in non-continuous $F$ or $G,$ but no new concepts are involved.)

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By definition, a copula is the joint distribution you get after you re-express the original variables in a particular way.

For continuous variables, as in this case, the re-expression is the Probability Integral Transform that replaces each possible value $x$ of a random variable $X$ (governed by a distribution $F$) by its quantile $u=F(x).$ You have already found the two quantile functions $$F,G:t\to \frac{1}{1 + e^{-t}}.\tag{1}$$

Let the re-expressed variables be $$U=F(X)\text{ and }V=G(Y).\tag{2}$$ The joint distribution of any pair of variables $(U,V)$ is

$$F_{(U,V)}(u,v) = \Pr(U \le u\text{ and } V \le v).\tag{3}$$

This is the copula. Note that this definition also means

$$ H(x,y) = \Pr(X\le x\text{ and } Y\le y)=\frac{1}{1 + e^{-x} + e^{-y}}.\tag{4}$$

Combining $(1),(2),(3)$ and simple algebraic manipulation of the descriptions of these events gives

$$\eqalign{F_{(U,V)}(u,v) &= \Pr(F(X) \le u\text{ and } F(Y) \le v)\\ &= \Pr\left(\frac{1}{1+e^{-X}} \le u \text{ and } \frac{1}{1+e^{-Y}} \le v\right) \\ &= \Pr\left(X \le \log\left(\frac{u}{1-u}\right) \text{ and } Y \le \log\left(\frac{v}{1-v}\right) \right) \\ &= H\left(\log\left(\frac{u}{1-u}\right), \log\left(\frac{v}{1-v}\right)\right). }$$

Plug this result into $(4)$ and do the algebra.