Copula Functions

Definition and Properties

In what follows, the definition of a copula is provided in the bivariate case.

A copula is a function C:[0,1] × [0,1] → [0,1] with the following properties:

• 1.
  For every u,v in [0,1], C(u, 0) = 0 = C(0,v), and C (u, 1) = u and C(1,v) = v.
• 2.
  For every u1, u2, ν1, V2 in [0,1], such that u1 ≤ u2 and v1 ≤ V2, C(u2, V2) – C(u2, ν1) – C(u1, V2) + C(u1,ν1) ≥ 0.

An example is the product copula

One important property of copulas is the Frécbet-Hoeffding bounds inequality, given by

where W and M are also copulas referred to as Fréchet-Hoeffding lower and upper bounds, respectively, and defined by W(u, v) = max (u + v − 1, 0) and M(u, v) = min(w, v).

Much of the usefulness of copulas in nonparametric statistics is due to the fact that for strictly monotone transformations of the random variables under interest, copulas are either invariant or [Page 257]change in predictable ways. Specifically, let X and Y be continuous random variables with copula Cxy, and let f and g be strictly monotone functions on Ran X and Ran Y (Ran: Range), respectively.

1. If f and g are strictly increasing, then

2. If f is strictly increasing and g is strictly decreasing, then

3. If f is strictly decreasing and g is strictly increasing, then

4. If f and g are strictly decreasing, then

Sklar's Theorem for Continuous Distributions

Sklar's theorem defines the role that copulas play in the relationship between multivariate distribution functions and their univariate margins.

Sklar's Theorem in the Bivariate Case

Let H be a joint distribution function with continuous margins F and G. Then there exists a unique copula C such that for all x, y∊

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