Statistics Multivariate analysis Theorem

Sklar's theorem

Builds on Copula Cumulative distribution function Marginal distribution Joint probability distribution

Statement

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

There is a copula $C$ such that for all $x_1, \ldots, x_d \in \R$ , $H(x_1, \ldots, x_d) = C(F_1(x_1), \ldots, F_d(x_d)).$
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.
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.

Going deeper

For a student

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.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Copula concept
Cumulative distribution function concept
Joint probability distribution concept
Marginal distribution concept
Sklar's theorem statement

Tag	`ZD7V7R27R9`
URI	`m2://ZD7V7R27R9`
Kind	statement, theorem
Writing	W0 stub
MSC	62H05, 60E05
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About	Copula
Uses	Copula, Cumulative distribution function, Marginal distribution, Joint probability distribution