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

Sklar's theorem is central to copula modelling. Which statement of its implication is correct?

Sklar's theorem says any joint distribution function can be expressed as a copula applied to its marginal distribution functions, with the copula unique for continuous marginals. This separates the dependence structure from the marginals, so normal marginals do not force a Gaussian copula.

  1. AAny joint distribution can be written as a copula applied to its marginal distribution functions, separating dependence from marginalsCorrect
  2. BA copula must always be Gaussian if the marginals are normal
  3. CThe copula is unique only if the marginals are discrete
  4. DThe copula determines the marginal distributions completely
  5. Linear correlation is fixed once the copula is chosen, regardless of marginals

Explanation

Sklar's theorem states F(x,y)=C(F_X(x),F_Y(y)), with C unique for continuous marginals. This separates dependence structure from marginals. Normal marginals do not force a Gaussian copula, and Pearson correlation still depends on the marginals.

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