Copulas and Dependence Beyond Marginals

Why Marginal Distributions Do Not Determine a Joint Law

Suppose two clinical measurements each have an exponential distribution. Those marginal distributions describe each measurement separately, but they do not say whether large values tend to occur together, whether one tends to be large when the other is small, or whether the measurements are independent. A copula records this missing part of the joint distribution: its dependence structure, separated from its margins.

This separation is useful when a marginal model is well understood but dependence remains uncertain. It also makes comparisons precise: we can hold the margins fixed and change the copula, then see how joint probabilities or other summaries change.

The two most familiar copulas are the independence copula \(C(u,v)=uv\) and the comonotone copula \(C(u,v)=\min(u,v)\). The former describes independent uniform variables; the latter describes two equal uniform variables. They have the same uniform margins, but very different joint behavior.

Sklar’s Representation

Let \(X\) and \(Y\) have joint cumulative distribution function \(H\), and marginal cumulative distribution functions \(F\) and \(G\). Sklar’s theorem identifies a copula that connects these distributions.

Here is a proof of the representation and uniqueness when the margins are continuous. This is the case used in the examples below. Define \(U=F(X)\) and \(V=G(Y)\). The probability integral transform says that \(U\) and \(V\) are each uniform on \([0,1]\). Their joint distribution function $$ C(u,v)=P(U\leq u,\ V\leq v) $$ is therefore a copula.

For continuous \(F\), the events \(\{X\leq x\}\) and \(\{F(X)\leq F(x)\}\) agree up to probability zero. Monotonicity gives one inclusion. Any discrepancy in the other direction can only occur where \(F\) is flat, and a flat interval has probability zero under the distribution \(F\). The same reasoning applies to \(G\). Consequently, $$ H(x,y)=P(F(X)\leq F(x),\ G(Y)\leq G(y))=C(F(x),G(y)). $$ This proves existence for continuous margins.

For uniqueness, let \(\widetilde C\) be another copula representing \(H\). For any \(u,v\) strictly between zero and one, continuity of \(F\) and \(G\) ensures that there are \(x,y\) with \(F(x)=u\) and \(G(y)=v\). Thus $$ \widetilde C(u,v)=H(x,y)=C(u,v). $$ Copulas are continuous: changing \(u\) by an amount \(\delta\) can change the probability by at most \(|\delta|\), because the first margin is uniform; the corresponding bound for changing \(v\) is \(|\delta|\). The equality therefore extends to the boundary of the unit square. This proves uniqueness. Conversely, composing a copula with \(F\) and \(G\) gives a joint distribution function: the rectangle increments are nonnegative because \(F\) and \(G\) are nondecreasing and \(C\) has nonnegative rectangle probabilities. Setting either argument to its marginal endpoint gives \(F\) and \(G\) as the resulting margins. \(\square\)

Worked Examples: Same Margins, Different Dependence

When a Copula Has a Density

If the joint distribution and margins are smooth, the copula also gives a convenient density formula. Suppose \(F\) and \(G\) are continuously differentiable with densities \(f\) and \(g\), and suppose the copula has a continuous density \(c(u,v)\), so that its mixed derivative is \(c\). Then the joint density is $$ h(x,y)=c(F(x),G(y))f(x)g(y). $$ To verify this, write \(H(x,y)=C(F(x),G(y))\). Differentiating first with respect to \(x\) and then with respect to \(y\), the chain rule gives $$ \frac{\partial^2 H(x,y)}{\partial y\,\partial x} =\frac{\partial^2 C(F(x),G(y))}{\partial v\,\partial u}f(x)g(y) =c(F(x),G(y))f(x)g(y). $$ Under the stated smoothness conditions, this mixed derivative is the joint density. For independence, \(c(u,v)=1\), and the formula reduces to \(h(x,y)=f(x)g(y)\). Not every copula has a density: the comonotone copula, for example, concentrates probability on the diagonal and is singular with respect to area.

Interpretation and a Common Pitfall

A copula describes how marginal ranks occur together. For continuous margins, \(F(X)\) and \(G(Y)\) are uniform, so a point such as \(C(0.9,0.9)\) is the probability that both variables fall at or below their respective 90th percentiles. This interpretation does not require the original measurements to share units or even to have similar marginal shapes.

A common mistake is to infer a joint distribution from the two marginal distributions alone. For example, knowing that two measurements are each unit-rate exponential does not justify multiplying their marginal probabilities. That multiplication gives the joint probability only if the variables are independent, which is the additional assumption \(C(u,v)=uv\). The independent, countermonotone, and mixture examples all have the same exponential margins but different probabilities of joint events.

Another useful implication is invariance under strictly increasing transformations: increasing a measurement’s scale or changing its units does not alter its rank, and hence does not alter its copula. A decreasing transformation reverses that variable’s ranks instead. This is why separating margins from dependence can help when comparing dependence across different measurement scales.