# Decomposing a random variable into marginals and copula

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!

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.)

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.

• Thank you for this. So the marginals are the actual quantile functions themselves? Also could you explain how the definition (3) implies (4) ? – gb4 May 27 '19 at 18:42
• $H$ is the (bivariate) distribution function--as you stipulate--so $(4)$ is merely a special case of $(3).$ – whuber May 27 '19 at 21:46