---
tag: ZD7V7R27R9
kind: statement
role: theorem
name: Sklar's theorem
msc: [62H05, "60E05"]
uses: [CAMJRBZANK, 0H8H7ZTHW2, 0MXT6PRZZ6, 0DR81R1F2B]
about: [CAMJRBZANK]
sources:
  - url: https://en.wikipedia.org/wiki/Copula_(statistics)
    locator: Sklar's theorem
---

## Statement

Let $d \ge 2$ and let $H$ be the joint [[cumulative distribution function|0H8H7ZTHW2]] of a random vector $(X_1, \ldots, X_d)$ with [[marginal|0MXT6PRZZ6]] distribution functions $F_1, \ldots, F_d$.

1. There is a [[copula|CAMJRBZANK]] $C$ such that for all $x_1, \ldots, x_d \in \R$,
$$H(x_1, \ldots, x_d) = C(F_1(x_1), \ldots, F_d(x_d)).$$
2. The copula $C$ is uniquely determined on $\operatorname{Ran} F_1 \times \cdots \times \operatorname{Ran} F_d$, where $\operatorname{Ran} F_i$ is the range of $F_i$.
   In particular $C$ is unique when $F_1, \ldots, F_d$ are continuous.
3. Conversely, if $C$ is a $d$-dimensional copula and $F_1, \ldots, F_d$ are distribution functions on $\R$, then $(x_1, \ldots, x_d) \mapsto C(F_1(x_1), \ldots, F_d(x_d))$ is a joint distribution function with marginals $F_1, \ldots, F_d$.

## Remarks

When the $F_i$ are continuous, the proof is short.
Each $U_i = F_i(X_i)$ is uniform on $[0, 1]$, and $X_i \le x_i$ holds exactly when $U_i \le F_i(x_i)$, up to an event of probability $0$.
So the joint distribution function $C$ of $(U_1, \ldots, U_d)$ satisfies part 1.
For part 3, take $U$ with distribution function $C$ and set $X_i = F_i^{-1}(U_i)$ with the generalised inverse $F_i^{-1}(u) = \inf \set{x : F_i(x) \ge u}$.
For discontinuous marginals, part 1 needs an extra randomisation or an extension argument.
Uniqueness then fails off the ranges: for two Bernoulli variables, many copulas give the same $H$.

Example: if $(X_1, X_2)$ is bivariate normal with correlation $\rho$, its copula is the Gaussian copula with parameter $\rho$.
Combining this copula with other marginals produces non-normal distributions with the same dependence.

## Explanations

### L1 human

Sklar's theorem says that a joint distribution is fully described by two separate pieces.
One piece is the list of the individual distributions of the coordinates.
The other piece, the copula, records only how the coordinates depend on each other.
The pieces can be chosen freely and glued back together.

### L2 human

The theorem justifies the two-step modelling used in practice: fit each marginal, then fit a copula to the ranks or to $F_i(X_i)$.
Many measures of dependence, such as Spearman's rho and Kendall's tau, depend only on the copula when the marginals are continuous.
The theorem also shows that the Fréchet–Hoeffding bounds for copulas give bounds for any joint distribution with given marginals.