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?
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.
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