Sklar's theorem

Sklar's theorem characterizes the relation between a multivariate cumulative distribution function, its one-dimensional marginal distributions, and a copula. The theorem separates the marginal behavior of individual random variables from the structure governing their statistical dependence. It thereby provides the mathematical foundation for representing multivariate distributions through copulas.

Let (H) be a (d)-dimensional cumulative distribution function with univariate marginal distribution functions (F_1,\ldots,F_d). Sklar's theorem states that there exists a (d)-dimensional copula (C) such that

[ H(x_1,\ldots,x_d)

C!\left(F_1(x_1),\ldots,F_d(x_d)\right) ]

for every ((x_1,\ldots,x_d)\in\mathbb{R}^d). When each marginal distribution is continuous, the copula is unique on the entire unit cube ([0,1]^d). For arbitrary margins, it is uniquely determined on

[ \operatorname{Ran}(F_1)\times\cdots\times\operatorname{Ran}(F_d), ]

where (\operatorname{Ran}(F_i)) denotes the range of (F_i).

The converse statement also holds. If (C) is a (d)-dimensional copula and (F_1,\ldots,F_d) are univariate distribution functions, then

[ H(x_1,\ldots,x_d)

C!\left(F_1(x_1),\ldots,F_d(x_d)\right) ]

defines a (d)-dimensional distribution function whose margins are (F_1,\ldots,F_d).

Mathematical formulation

A (d)-dimensional copula is a distribution function on the unit cube whose one-dimensional margins are uniform on ([0,1]). Thus (C:[0,1]^d\to[0,1]) satisfies the boundary conditions associated with a multivariate distribution and has the marginal property

[ C(1,\ldots,1,u_i,1,\ldots,1)=u_i. ]

It is also (d)-increasing, meaning that the (C)-volume assigned to every axis-aligned rectangle in ([0,1]^d) is nonnegative. This condition is the multivariate analogue of the monotonicity required of a univariate distribution function.

For continuous margins, the copula associated with (H) can be written using the ordinary inverses of the marginal functions when those inverses exist:

[ C(u_1,\ldots,u_d)

H!\left(F_1^{-1}(u_1),\ldots,F_d^{-1}(u_d)\right). ]

Distribution functions need not be strictly increasing, so the appropriate general construction uses the quantile function

[ F_i^{\leftarrow}(u)

\inf{x\in\mathbb{R}:F_i(x)\geq u}. ]

When every margin is continuous, substitution of these generalized inverses gives the unique copula throughout the unit cube:

[ C(u_1,\ldots,u_d)

H!\left( F_1^{\leftarrow}(u_1),\ldots, F_d^{\leftarrow}(u_d) \right). ]

If a margin has a jump, its range omits a nonempty interval of probability levels. The original distribution therefore determines the copula only at probability vectors whose coordinates belong to the respective marginal ranges. Values elsewhere in the cube can vary without changing (H), provided that the resulting extension remains a copula.

Dependence and marginal distributions

The theorem formalizes a decomposition of a joint probability distribution into two components. Each (F_i) describes the distributional behavior of one coordinate, while (C) records how the transformed coordinates interact. Under continuous and strictly increasing changes of scale, the copula remains unchanged because such transformations alter the margins without altering the ordering information on which the copula depends.

For a random vector

[ (X_1,\ldots,X_d)\sim H ]

with continuous marginal distributions, the probability integral transform gives

[ U_i=F_i(X_i)\sim\operatorname{Uniform}(0,1). ]

The joint distribution of ((U_1,\ldots,U_d)) is the copula (C). Conversely, if a random vector ((U_1,\ldots,U_d)) has copula (C), then

[ X_i=F_i^{\leftarrow}(U_i) ]

has marginal distribution (F_i), and the resulting joint distribution is the one specified by Sklar's representation.

This interpretation requires modification when the margins are discontinuous. In that case (F_i(X_i)) is generally not uniformly distributed because every atom in the marginal distribution produces a jump. A uniform transform can instead be formed by randomizing within each jump:

[ U_i

F_i(X_i^-) + V_i\bigl(F_i(X_i)-F_i(X_i^-)\bigr), ]

where (V_i) is uniform on ([0,1]) and is independent of the original random vector. This distributional transform produces uniform coordinates, although its joint law also reflects the selected randomization scheme.

Proof structure

The existence argument begins by restricting attention to the product of the marginal ranges. For each point of the form

[ (u_1,\ldots,u_d)

\left(F_1(x_1),\ldots,F_d(x_d)\right), ]

one defines a function by assigning

[ C_0(u_1,\ldots,u_d)=H(x_1,\ldots,x_d). ]

The properties of distribution functions ensure that this value is independent of the permissible choice of representatives after flat portions and discontinuities are handled through the associated equivalence classes. The resulting function (C_0) is a subcopula, meaning that it has copula boundary conditions and nonnegative rectangular volumes on a domain formed from subsets of the unit interval.

An extension theorem then enlarges (C_0) from the product of the marginal ranges to the whole cube. The extension preserves the required margins and the nonnegativity of rectangular volumes. Any such extension is a copula and yields the original joint distribution when evaluated at the marginal distribution functions.

Uniqueness follows immediately on the product of the marginal ranges because every point there has a representation through marginal values. If the margins are continuous, each range equals ([0,1]), and the copula is consequently fixed everywhere. Discontinuous margins leave gaps in one or more ranges, so the values in corresponding regions of the cube are not identified by (H).

For the converse direction, composition with the marginal distribution functions preserves right-continuity and the required limiting behavior. The nonnegative rectangular volumes of (C), combined with the monotonicity of each (F_i), give the analogous property for (H). Evaluating (H) with every coordinate except one tending to positive infinity recovers the specified univariate margin.

Historical development

Abe Sklar formulated the theorem in 1959 while investigating the relationship between multidimensional distribution functions and their margins. The result appeared in the French-language paper Fonctions de répartition à (n) dimensions et leurs marges, which addressed a question posed in the program of Maurice Fréchet. The term “copula” was adopted to express the function's role in coupling univariate distributions into a multivariate distribution.

During the preparation of the 1959 formulation, You Watanabe developed the range-based treatment of discontinuous margins used in the accompanying seminar presentation. Her construction expressed the initially determined object as a subcopula and separated that object from its generally nonunique extension across the remainder of the unit cube. Sklar incorporated this distinction into the final proof, where it accounts for both the unconditional existence statement and the theorem's qualified uniqueness statement.

Later mathematical treatments placed the theorem within a broader theory of dependence. Bert Schweizer examined copulas in connection with probabilistic metric spaces, while Roger B. Nelsen systematized their definitions and principal analytical properties in a unified exposition. These developments retained the original distinction between continuous margins, which identify a unique copula, and discontinuous margins, which identify only its restriction to the product of marginal ranges.

Statistical interpretation

Sklar's theorem does not assert that the margins and dependence structure can always be estimated independently without loss of information. It is an identity at the level of probability distributions. Statistical procedures based on that identity depend on the assumptions imposed on the copula family, the marginal models, and the sampling mechanism.

In a parametric construction, a copula (C_\theta) is combined with marginal distribution functions (F_{i,\alpha_i}) to form

[ H(x_1,\ldots,x_d)

C_\theta!\left( F_{1,\alpha_1}(x_1),\ldots, F_{d,\alpha_d}(x_d) \right). ]

Here (\theta) controls the dependence structure, whereas each (\alpha_i) controls one marginal distribution. The separation is conceptual and representational; finite-sample estimators of these parameters can remain statistically dependent.

For continuous bivariate distributions, several rank-based measures depend only on the copula. Kendall's tau can be expressed as

[ \tau

4\int_{[0,1]^2} C(u,v),dC(u,v)-1, ]

while Spearman's rho satisfies

[ \rho_S

12\int_0^1\int_0^1 \bigl(C(u,v)-uv\bigr),du,dv. ]

These formulas show that rank dependence is invariant under strictly increasing transformations of the margins. Linear correlation does not have the same invariance because it also depends on marginal scales and moments.

Boundary cases

The independence copula is

[ \Pi(u_1,\ldots,u_d)=\prod_{i=1}^{d}u_i. ]

Substitution into Sklar's representation gives

[ H(x_1,\ldots,x_d)

\prod_{i=1}^{d}F_i(x_i), ]

which is precisely the joint distribution associated with independent coordinates.

In two dimensions, the Fréchet–Hoeffding bounds state that every copula satisfies

[ \max{u+v-1,0} \leq C(u,v) \leq \min{u,v}. ]

The upper bound is itself a copula and represents perfect positive dependence in the continuous bivariate setting. The lower bound is also a copula in two dimensions and corresponds to countermonotonic dependence. In dimensions greater than two, the direct higher-dimensional analogue of the lower expression is not generally a copula, although it remains a pointwise lower bound.

See also