Copula Definition and Sklar's Theorem Explained Simply

Understand Copula Definition and Sklar's Theorem

Key rules to remember

Definition of a copula

C(u, v) = P(U ≤ u, V ≤ v), with U, V ~ Uniform(0,1)

C is defined on [0,1] × [0,1]. Both marginals must be uniform.

Boundary conditions

C(u, 0) = C(0, v) = 0; C(u, 1) = u; C(1, v) = v

These come from uniform marginals. They are used to check whether a given function is a copula.

2-increasing property

C(u2, v2) − C(u1, v2) − C(u2, v1) + C(u1, v1) ≥ 0 for u1 ≤ u2, v1 ≤ v2

This ensures rectangles have non-negative probability.

Sklar's theorem

F(x, y) = C(F_X(x), F_Y(y))

C is unique if the marginals are continuous.

Inverse form of Sklar

C(u, v) = F(F_X⁻¹(u), F_Y⁻¹(v))

Valid for continuous marginals. Use it to extract the copula from a joint distribution.

Independence copula

C(u, v) = uv

X and Y are independent exactly when their copula is uv.

Fréchet–Hoeffding bounds

max(u + v − 1, 0) ≤ C(u, v) ≤ min(u, v)

Every copula lies between these bounds. The lower bound is the countermonotonic copula W and the upper bound is the comonotonic copula M.

Survival probability from a copula

P(X > x, Y > y) = 1 − F_X(x) − F_Y(y) + C(F_X(x), F_Y(y))

Use inclusion–exclusion with u = F_X(x) and v = F_Y(y).

How to solve Copula Definition and Sklar's Theorem questions

1. 1Identify what you are given: a joint distribution, a copula, or the marginals. Note whether the marginals are continuous.
2. 2Convert each variable to a uniform value: u = F_X(x) and v = F_Y(y). Compute these numbers first.
3. 3If asked whether a function is a copula, check the boundary conditions (C(u,0) = 0, C(u,1) = u, and so on) and then the 2-increasing property.
4. 4Apply Sklar's theorem: joint probability P(X ≤ x, Y ≤ y) = C(u, v). Substitute the values.
5. 5For other regions, use inclusion–exclusion. For example P(X > x, Y > y) = 1 − u − v + C(u, v).
6. 6To extract a copula from a joint distribution, invert the marginals: C(u, v) = F(F_X⁻¹(u), F_Y⁻¹(v)). Simplify.
7. 7Check your answer against the Fréchet bounds: it must lie between max(u + v − 1, 0) and min(u, v).
8. 8State assumptions, such as continuous marginals, and give the final numerical result clearly.

Quickest way: Transform, plug in, check bounds

Common mistakes in Copula Definition and Sklar's Theorem

• Thinking the copula contains the marginal distributions.

  The joint distribution F looks like one object, so students assume C holds everything.

  Fix: Remember that C only describes dependence. The marginals are fed into C through u = F_X(x) and v = F_Y(y).

• Putting x and y directly into C instead of F_X(x) and F_Y(y).

  Students forget that C is defined on [0,1] × [0,1].

  Fix: Always convert to uniforms first. Write u = ... and v = ... as the first line of working.

• Computing P(X > x, Y > y) as 1 − C(u, v).

  It is a tempting complement rule, but the complement of 'both below' is not 'both above'.

  Fix: Use inclusion–exclusion: P(X > x, Y > y) = 1 − u − v + C(u, v).

• Checking only the 2-increasing property, or only the boundary conditions, when testing for a copula.

  Students stop once one condition holds.

  Fix: A copula needs both the boundary conditions and the 2-increasing property. Check boundaries first because it is faster.

• Saying the copula is always unique.

  Sklar's theorem is remembered without its condition.

  Fix: State that uniqueness holds when the marginals are continuous. With discrete marginals, several copulas can give the same joint distribution.

• Treating the Fréchet bounds as copulas that apply to any variables regardless of the marginals.

  The bounds are written in u and v, so students forget the transformation.

  Fix: Apply the bounds to u = F_X(x) and v = F_Y(y). The upper bound min(u, v) corresponds to comonotonic variables and the lower bound max(u + v − 1, 0) to countermonotonic variables.

Worked examples