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

For two continuous random variables, the Fréchet–Hoeffding upper bound copula M(u,v)=min(u,v) corresponds to which situation?

The upper Fréchet–Hoeffding bound min(u,v) is the comonotonic copula, arising when one variable is a strictly increasing function of the other, so they have perfect positive dependence. Independence is uv, and the lower bound arises under countermonotonicity.

  1. AX and Y independent
  2. BY is a strictly increasing function of X (comonotonic)Correct
  3. CY is a strictly decreasing function of X (countermonotonic)
  4. DX and Y are jointly normal with correlation zero
  5. X and Y have Gumbel tail dependence of exactly 0.5

Explanation

Every copula satisfies max(u+v−1,0) ≤ C(u,v) ≤ min(u,v). The upper bound is attained when Y is an increasing function of X, i.e. perfect positive dependence. The lower bound corresponds to a decreasing function, and independence lies strictly between.

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