Help understanding uniform marginal distribution in Farlie-Morgenstern family.

http://imgur.com/FeFf3e9

The imgur link is to a screenshot of the relevant section in my text. I have trouble understanding how if $H(x, \infty)=F(x)$ is the marginal distribution of $x$, how $F(x) = x, 0 < x < 1$ is the uniform marginal. How can the marginal distribution be uniform if it equals $F(x)=x$? Furthermore how can a CDF function be uniform?

What does "uniform" refer to in this text?

• If at all possible, I would recommend simply putting the relevant text from your image directly in your question, and cite the source. – Patrick Coulombe Mar 5 '14 at 0:05

Uniform margins are true of every copula, not just FGM copulas (not sure why that book leaves Gumbel out of the usual copula name).

In this case, uniform means "Has a uniform distribution" - that is, the density is constant over some interval, and in the case of copulas, where that interval is $[0,1]$, the corresponding distribution function is of the form $F_X(x)=x$ over that interval (and is 0 to the left of it and 1 to the right of it). Note that the cdf, $F_X(x)=x$ corresponds to a constant density, $f$, which is why the distribution is called 'uniform'.

Note that copulas have uniform $[0,1]$ marginals by definition. The particular copula you refer to has been chosen to fit with the definition. That it does so is easily seen; substitute $y=1$ into $H$ to see it for $X$, and $x=1$ into $H$ to see it for $Y$. Alternatively you can compute the marginals from the joint density by integration.

A search here for copulas will get you a lot more basic information relating to copulas.

• Thanks for your answer, but I'm confused if the marginals have a uniform distribution, how can Fx(x)=x? How is that uniform, wouldn't the distribution depend on x and not just be constant? – sayyid monroe Mar 5 '14 at 0:12
• I don't understand your question. ... unless you're confused about the difference between density and distribution perhaps. In any case, I am really not sure what you're asking. The density $f$, for $F(x)=x$ is constant (i.e. uniform). Did you read the link? – Glen_b Mar 5 '14 at 0:37
• I guess my question does concern the difference between the density and distribution. I calculated the density by taking the integral of the joint density and found that it is indeed 1 and thus uniform. However I don't understand the statement that the marginal distributions are H(x,∞) = F(x). F(x)=x and that is not uniform. – sayyid monroe Mar 5 '14 at 1:19
• integrate the marginal density $(\,f_X(x)=1,\quad 0\leq x\leq 1;\,0\text{ elsewhere })$, and you obtain the marginal distribution function, $F_X(x)=x, \quad 0\leq x\leq 1;\,=0,\quad x<0;\,=1,\quad x>1$, or equivalently, differentiate $F$ to get $f$. If you're not familiar with these basic relationships, you should not be trying to understand copulas yet, because you don't have the necessary tools. – Glen_b Mar 5 '14 at 1:29
• Ok. Thanks for walking me through it all; that was actually very simple. The fact that FX(x)=x=F(x) really messed me up as I kept getting hung up on the fact that F(x) is a CDF of X and didn't understand how that equaled the marginal distribution of X. – sayyid monroe Mar 5 '14 at 1:35