Module 18: Copulas Flashcards
Copula
A copula is a joint cumulative distribution function expressed in terms of the marginal cumulative distribution functions.
A copula in n-dimensions would be expressed as:
C(u) = C(u₁, u₂, …, uₙ) = P(U₁ ≤ u₁, U₂ ≤ u₂, …, Uₙ ≤ uₙ)
Where uᵢ = Fₓᵢ(𝑥ᵢ), ie the values of the individual cumulative distribution functions (CDFs), each of which lie in the range [0,1].
A copula is determined by the relative order of the observations rather than by the exact shape of the marginal distribution (the invariance property).
A copula must have 3 properties
- It must be an increasing function of its inputs.
- If the values of all but one of the marginal CDFs are equal to 1, then the copula is equal to the value of the remaining marginal CDF.
- The copula must always return a non-negative probability.
Sklar’s Theorem
Say’s that if F is a joint CDF and F₁, …, Fₙ are marginal CDFs, then there exists a copula such that for all 𝑥₁, …, 𝑥ₙ ϵ [-∞, ∞]
F_{X₁, …, Xₙ} ( 𝑥₁, …, 𝑥ₙ) = C_{X₁, …, Xₙ} [ F_𝑥₁ (𝑥₁), …, F_𝑥ₙ (𝑥ₙ) ]
Furthermore if the marginal distributions are continuous, then the copula is unique.
Survival Copula
For each copula, there is a corresponding survival copula expressing the joint survival probability in terms of the marginal survival probabilities.
Concordance (or association)
Concordance between two variables, X and Y, does not necessarily imply dependence (eg the two variables may instead be linked by a third variable).
3 Measures of concordance
- Pearson’s rho
- Spearman’s rho
- Kendall’s tau
Explicit copulas
have simple closed-form expressions.
An important subclass is that of Archimedian copulas
4 Examples of Archimedian copulas
- Gumbel
- Frank
- Clayton
- Generalised Clayton
Archimedian copulas
Define closed-form probability distributions, but are limited by the small number of parameters available to describe multivariate relationships.
Implicit Copulas
Based on well-known multivariate distributions, but no simple closed-form expression exists for them. Examples are the Normal (or Gaussian) copula and the (Student’s) t-copula.
The t-copula allows for more flexibility than the normal copula in the level of tail dependency. The smaller the value of γ, the greater the level of tail dependency. As γ →∞, the t-copula tends to the normal copula.
Choosing copulas
Possible copula candidates might be selected based on features of the sample data:
- upper, lower (and other) ‘tail’ dependencies
- patterns of dependence, eg general positive co-dependence.
Fitting copulas
Copulas might be fitted using::
- maximum likelihood
- parameterisation based on rank correlations in the sample