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 F_X and F_Y, there exists a copula C such that F(x,y) = C(F_X(x), F_Y(y)). Which statement about uniqueness is correct?
The copula is uniquely determined only on the product of the ranges of the marginal distribution functions. When both marginals are continuous, those ranges cover [0,1], so the copula is unique. For discrete marginals it need not be.
- AC is always unique for any marginals
- BC is never unique
- CC is unique on the range of F_X multiplied by the range of F_Y, so it is unique when both marginals are continuousCorrect
- DC is unique only if X and Y are independent
- C is unique only if the marginals are identical
Explanation
Sklar's theorem guarantees existence of C. C is uniquely determined on Ran(F_X) x Ran(F_Y). With continuous marginals the ranges are the whole of [0,1], so C is unique. With discrete marginals many copulas fit the same F.
Did you get it right without looking?
One question tells you little. A timed set on Introduction to copulas shows your real accuracy, how long you take and where you lose marks.
More Introduction to copulas questions
- Which statement about Archimedean copulas is correct?
- An insurer finds the empirical probability P(U1 > 0.99 | U2 > 0.99) is about 0.60 and P(U1 > 0.999 | U2 > 0.999) is about 0.05 for pseudo-ob…
- Two loss variables X and Y are linked by a Clayton copula with parameter α = 2, where C(u,v) = (u^(-α) + v^(-α) − 1)^(-1/α). The lower tail …
- For two continuous random variables, the Fréchet–Hoeffding upper bound copula M(u,v)=min(u,v) corresponds to which situation?
- A risk manager at an Indian general insurer models flood and cyclone losses and finds they tend to be extreme together far more often than a…
- A bank uses a Gaussian copula to model joint defaults across a loan portfolio. After a crisis, it is criticised because the model understate…