Marginals are mentioned a lot in copula literature, what does the term really mean?
For example what is the intuitive meaning behind a statement like
"This function describes the dependence structure separated from the marginals."
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2$\begingroup$ "Yes". The problem with asking questions to which the answer is simply yes or no is we can't actually give a single word as an answer, so the question may technically remain unanswered. Please try to rephrase in a way that allows a more extensive answer, useful to later readers. One good choice would be "Marginals are mentioned a lot in copula literature, what does the term mean?". Even though you did correctly understand already, it allows people to give an actual answer - one that's useful to others. Please consider editing your question. $\endgroup$– Glen_bCommented Jan 30, 2015 at 10:51
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$\begingroup$ Adding the quote is a nice touch. $\endgroup$– Glen_bCommented Jan 30, 2015 at 16:41
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1$\begingroup$ The "marginal [distributions]" are what you get when you focus on a single variable and simply ignore the rest. $\endgroup$– whuber ♦Commented Jan 30, 2015 at 17:16
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2 Answers
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The term 'marginals' is a loose reference to marginal distributions (as you originally supposed).
That's certainly what the quoted sentence is discussing.
More specifically, without any other adjective, the unqualified term "marginals" would refer to the univariate marginal distributions of each of the variables that are related by a copula.
In the case of copulas, two different multivariate distributions can have the same "dependence structure" as measured by a copula. After a univariate transformation of each of the (univariate) random variables involved in the multivariate distribution to uniformity, the joint distribution that results is called a copula. If you then transform each of the individual random variables (changing the "marginals"), the copula is unchanged, even though the multivariate distribution is changed.
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$\begingroup$ It's a nice answer, but it strikes me that it will appear perfectly circular to anyone who does not already know the definition of "marginal distribution"; and that might be what is at issue here. $\endgroup$– whuber ♦Commented Jan 30, 2015 at 17:17
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$\begingroup$ I'm not sure there's a difference but the context in which "marginals" is often used is in referring to the cumulative distribution functions of the individual random variables themselves. Which is not quite the definition on the wikipedia page of marginal distributions, hence my slight confusion. $\endgroup$– Henry ECommented Jan 30, 2015 at 19:47
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1$\begingroup$ Henry -- Strictly speaking "marginal distribution" is the CDF you mention. Often people loosely say 'distribution' when they mean 'density' ... and mostly that looseness causes no problem since it's clear from context. So it's quite correct to refer to those CDFs as 'marginal distributions'. They are! $\endgroup$– Glen_bCommented Jan 31, 2015 at 0:09
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Marginals are refering to the probability distributions of individual random variables, typically the cummulative distribution function.
I think it seems counter intuitive at first because in most examples you start by looking at the marginals and not the collection, without which we would not be refering to the distributions of the individuals random variables as marginals.