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

Sklar's theorem states that for a joint distribution function F with marginals F1 and F2, there exists a copula C such that:

Sklar's theorem says any joint distribution function can be written as a copula applied to its marginal distribution functions, F(x,y)=C(F1(x),F2(y)). The copula always exists and is unique when the marginals are continuous.

  1. AF(x,y) = C(F1(x), F2(y)) for all x and yCorrect
  2. BF(x,y) = C(x,y) only when X and Y are independent
  3. CF(x,y) = F1(x) + F2(y) − C(x,y)
  4. DF(x,y) = C(F1(x), F2(y)) only when the marginals are continuous, and no copula exists otherwise
  5. F(x,y) = C(F1(x)F2(y), 1)

Explanation

Sklar's theorem gives F(x,y)=C(F1(x),F2(y)) for some copula C always. When the marginals are continuous, C is unique. Option 3 is wrong because existence does not need continuity, only uniqueness does.

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