2 Copulae Theory

Notation used in this Chapter is based on the book by Nelsen (Nelsen 2006).

First, let’s define a distribution function: a distribution function is a function \(F\) with domain \(\mathbf{R}\) such that

  • \(F\) is non-decreasing;

  • \(F(-\infty) = 0\) and \(F(+\infty) = 1\).

For a continuous random variable \(X\), its distribution function is defined as \(F_X(x) = P(X \le x)\).

Analogously, a joint distribution function is a function \(H\) with domain \(\mathbf{R}^2\) such that

  • \(H\) is 2-increasing;

  • \(H(x, -\infty) = H(-\infty, y) = 0\) and \(F(+\infty, +\infty) = 1\).

It can be showed that \(H\) has margins \(F(x) = H(x, +\infty)\) and \(G(y) = H(+\infty, y)\).

Formally, a two-dimensional copula is a function \(C\) with the following properties:

  • The domain of \(C\) is \(\mathbf{I}^2\);

  • For every \(u\), \(v\) in \(\mathbf{I}\): \[ C(u, 0) = 0 = C(0, v) \] and \[ C(u, 1) = u, \ C(1, v) = v \]

  • For every \(u_1\), \(u_2\), \(v_1\), \(v_2\) in \(\mathbf{I}\) such that \(u_1 \le u_2\), \(v_1 \le v_2\): \[ C(u_2, v_2) - C(u_2, v_1) - C(u_1, v_2) + C(u_1, v_1) \ge 0 \]

\(C(u, v)\) can be thought of as an assignment of a number in \(\mathbf{I}\) to the rectangle \([0, u] \times [0, v]\).

Finally, partial derivatives of a copula exist for any \(u\), \(v\) in \(\mathbf{I}\) and for almost all \(v\), \(u\) (respectively) such that: \[ 0 \le \frac{\partial C(u, v)}{\partial u} \le 1 \] \[ 0 \le \frac{\partial C(u, v)}{\partial v} \le 1 \] The existence of the partial derivatives is immediate because monotone, continuous functions (the margins e.g. \(F\), \(G\)) are differentiable almost everywhere.

2.1 Sklar’s Theorem

The fundamental theorem underlying copulae theory is Sklar’s theorem (Sklar 1959).

Let \(H\) be a joint distribution function with margins \(F\) and \(G\). Then, there exists a copula function \(C\) such that for all \(x\), \(y\) in \(\mathbf{R}\): \[ H(x, y) = C(F(x), G(y)) \] If \(F\) and \(G\) are continuous, then \(C\) is unique; otherwise, \(C\) is uniquely determined on \(\text{Ran} F \times \text{Ran} G\) (or either, when only one of the two is not continuous).

Conversely, if \(C\) is a copula and \(F\) and \(G\) are distribution functions, then \(H\) (as defined above) is a joint distribution function with margins \(F\) and \(G\).

Finally, Sklar’s theorem still holds when \(X\) and \(Y\) are random variables. If \(X\) and \(Y\) are random variables with distribution functions \(F_X\) and \(G_Y\), there exists a copula function \(C\) such that the joint distribution \(H_{X,Y}\) is uniquely determined (when \(X\) and \(Y\) are continuous) by \(C_{X,Y}(F_X, G_Y)\): \[ H_{X, Y} = C_{X, Y}(F_X, G_Y) \] If \(X\) and \(Y\) are discrete random variables, once again, \(C\) is defined on \(\text{Ran} X \times \text{Ran} Y\).

2.2 Product Copula

The product copula is defined as \(\Pi(u, v) = u v\). This always applies when the two marginal distributions are independent: \[ H(x, y) = \Pi(F_X(x), G_Y(y)) = F_X(x) G_Y(y) \leftrightarrow X \perp Y \]

2.3 Fréchet-Hoeffding Bounds

For every copula \(C\) and every \((u, v)\) in \(\mathbf{I}^2\), it can be showed that \[ W(u, v) \le C(u, v) \le M(u, v) \] where \(W(u, v) = \max(u + v - 1, 0)\) and \(M(u, v) = \min(u, v)\) are commonly referred to as the Fréchet-Hoeffding lower and upper bound, respectively.

References

Nelsen, Roger B. 2006. An Introduction to Copulas. Springer-Verlag New York. https://doi.org/10.1007/0-387-28678-0.

Sklar, A. 1959. “Fonctions de Répartition à N Dimensions et Leurs Marges.” Publications de L’Institut de Statistique de L’Université de Paris 8: 229–31.