Copulas as a Link Between Marginal and Joint Distributions
Summary
The document asks how copulas connect known marginal distributions to a joint distribution when random variables are dependent. It identifies the joint cumulative distribution as distinct from the product of the marginals, which applies only under independence, and proposes that a copula combines marginal cumulative probabilities to recover the joint cumulative probability. It also describes transforming variables through their marginal distributions into uniform variables, then mapping back to the original scales.
The discussion raises questions about whether this representation is one-to-one and how to interpret it geometrically for continuous distributions. It does not include an answer or establish the exact conditions of Sklar’s theorem. In particular, it blurs a copula’s role in representing dependence with the generation of samples and with contours of a joint density. The document is therefore useful as a statement of the core modeling intuition and the conceptual distinctions that need clarification, rather than as a complete derivation or implementation guide.
Key ideas
- A joint distribution generally cannot be recovered by multiplying dependent variables’ marginal distributions.
- A copula combines marginal cumulative probabilities to represent the joint cumulative distribution.
- Transforming observations through continuous marginal CDFs produces uniform variables whose dependence is captured by the copula.
- The document asks whether the mapping is one-to-one but does not resolve the conditions or implications.
- A copula represents dependence and should not be confused with a direct map to density contours or a sample generator.
Tags
Full text
# trying to better understand copulas # trying to better understand copulas This topic is dense with notation that makes things a bit confusing. But is this the correct interpretation? Suppose we have two jointly distributed random variables – $X$ and $Y$ – of arbitrary (but let's assume known) CDFs. The problem is the joint probability for any pair of values (x,y) is not simply $F_X(x)F_Y(y)$ because they are not independent. That actual joint distribution is what it is, and seems to be often called $H(x,y)$ in this literature. Now, it seems to me that the copula, in the end, is simply a function such that $C(F_X(x),F_Y(y))$ actually maps to the value for $H(x,y)$. It accomplishes this by sort of running the marginals "backwards". Everyone knows how to generate normal random samples in Excel from the $U[0,1]$ it provides by using the inverse normal function. But, with copulae, no matter what the marginals, you invert so you are looking at a d-dimensional distribution with uniform distributions. So, in the end, is the Copula just a mapping in the $[0,1]^d$ space where all you need give it (once you have it calculated) is the naïve values for the marginals of $F_X(x)$ and $F_Y(y)$, and it delivers up a pair of values in the $[0,1]^d$ space that, once transformed back into the actual original space give $H(x,y)$? In other words, Sklar's theorem guarantees there is a one-to-one mapping between $F_X(x)$ and $F_Y(y)$ to $H(x,y)$, and the copula captures all that information in a mapping in the uniform marginal space? And, need it even be 1:1? For, if we imagine a bivariate joint normal distribution (or any bivariate distribution of two continuous random variables), then it seems there is always a single $u$ in a 'uniform' space, that when run the other way gives you the right number for the joint probability of the bivariate distributions we start with (corresponding to contours of equal likelihood on the joint PDF)?
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.