Module 18: Copulas Flashcards

1
Q

Copula

A

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).

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2
Q

A copula must have 3 properties

A
  • 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.
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3
Q

Sklar’s Theorem

A

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.

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4
Q

Survival Copula

A

For each copula, there is a corresponding survival copula expressing the joint survival probability in terms of the marginal survival probabilities.

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5
Q

Concordance (or association)

A

Concordance between two variables, X and Y, does not necessarily imply dependence (eg the two variables may instead be linked by a third variable).

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6
Q

3 Measures of concordance

A
  • Pearson’s rho
  • Spearman’s rho
  • Kendall’s tau
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7
Q

Explicit copulas

A

have simple closed-form expressions.

An important subclass is that of Archimedian copulas

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8
Q

4 Examples of Archimedian copulas

A
  1. Gumbel
  2. Frank
  3. Clayton
  4. Generalised Clayton
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9
Q

Archimedian copulas

A

Define closed-form probability distributions, but are limited by the small number of parameters available to describe multivariate relationships.

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10
Q

Implicit Copulas

A

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.

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11
Q

Choosing copulas

A

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.
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12
Q

Fitting copulas

A

Copulas might be fitted using::

  • maximum likelihood
  • parameterisation based on rank correlations in the sample
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