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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to copulas

A bivariate copula C(u,v) is a function defined on the unit square. Which of the following statements is a defining property of a copula?

A copula is a joint distribution function on the unit square with uniform marginals. So it equals zero when either argument is zero, and C(u,1)=u and C(1,v)=v. It need not be the product copula, a density, or symmetric.

  1. AIts marginal distributions are exponential with mean 1
  2. BC(u,0) = 0 = C(0,v), and C(u,1) = u and C(1,v) = vCorrect
  3. CC(u,v) is always equal to the product uv
  4. DC(u,v) is a probability density that integrates to 1 over the unit square
  5. C(u,v) is symmetric in u and v for every copula

Explanation

A copula is a joint distribution function on [0,1]^2 with uniform(0,1) marginals. This requires C(u,0)=C(0,v)=0 and C(u,1)=u, C(1,v)=v, together with the 2-increasing property. The product uv is only the independence copula, and symmetry is not required of every copula.

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