What copulas are for

Sklar's theorem. Every joint distribution splits into its margins and a copula, and any margins and any copula recombine into a valid joint distribution:

\[H(x_1, \dots, x_d) = C\bigl(F_1(x_1), \dots, F_d(x_d)\bigr).\]

That is the whole practical appeal. Fit each margin however suits it — a fitted Gamma for claim sizes, a Student-t for returns, a kernel estimate for something awkward — and choose the dependence structure separately.

import rcopula as rc
from scipy import stats

joint = rc.CopulaDistribution(
    rc.ClaytonCopula(2.0),
    margins=[stats.norm(loc=1, scale=2), stats.expon(scale=1 / 3)],
)
x = joint.rvs(4000, random_state=0)

Going the other way needs no knowledge of the margins at all — only ranks:

u = rc.pseudo_obs(x)
rc.fit(rc.ClaytonCopula(), u, method="mpl")

Why rank correlation, not Pearson

Rank measures are invariant under increasing transforms of the margins. Squaring one variable and exponentiating another leaves Kendall's τ unchanged to machine precision, and moves Pearson's correlation substantially. Since the choice of marginal scale is usually arbitrary, a dependence measure that responds to it is measuring the wrong thing.

The narrower lesson matters too: no scalar summarises dependence. For \(y = x^2\) with symmetric \(x\), the two are perfectly dependent — knowing \(x\) determines \(y\) — and Pearson, Kendall and Spearman all report approximately zero, because the relationship is not monotone.

Tail dependence

\[\lambda_L = \lim_{u \downarrow 0}\frac{C(u,u)}{u},\qquad \lambda_U = \lim_{u \uparrow 1}\frac{1 - 2u + C(u,u)}{1-u}.\]

A non-zero value means joint extremes occur with a probability that does not vanish relative to marginal extremes. This is the property that separates families calibrated to the same τ, and it is the property that decides risk numbers.

The Gaussian copula's zero is not reassuring

Its tail dependence is exactly zero, and it reaches that limit logarithmically. At τ = 0.5 the tail concentration is still 0.27 at the 1-in-100 level and 0.10 at 1-in-10,000, while a Student-t at the same τ has settled to its limit by 1-in-100. "No tail dependence" describes behaviour at quantiles nobody observes; at the levels a risk report uses, the Gaussian copula looks substantially tail-dependent.

rc.plots.tail_concentration(u, [c1, c2]) draws exactly this.

The Kendall distribution function

For a copula \(C\) and \(\mathbf U \sim C\), \(K(t) = P(C(\mathbf U) \le t)\) is the univariate summary of a multivariate dependence structure. It determines Kendall's τ through \(\tau = 3 - 4\int_0^1 K\), and it answers a question a joint quantile cannot: how often is an event this severe, when "severe" is a region rather than a point.

The Kendall return period \(1/(1 - K(t))\) is the multivariate analogue of a return period, and it is never shorter than the univariate one.