Other Copulas

Some copulas, while necessary in certain cases and very useful, are hard to classify. We gather them here for simplicity.

Independence and Fréchet-Hoeffding bounds

IndependentCopula

Copulas.IndependentCopula Type

IndependentCopula(d)

The independent copula in dimension $d$ has distribution function

$$C(\mathbf{x})=\prod _{i=1}^d{x}_i.$$

It is Archimedean with generator $\psi (s)=e^{-s}$.

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

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MCopula

Copulas.MCopula Type

MCopula(d)

The upper Fréchet–Hoeffding bound is the copula with the largest value among all copulas; it corresponds to comonotone random vectors. For any copula $C$ and all $\mathbf{u}\in [0,1{]}^d$,

$$W(\mathbf{u})\le C(\mathbf{u})\le M(\mathbf{u}).$$

Both Fréchet–Hoeffding bounds are Archimedean copulas.

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

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WCopula

Copulas.WCopula Type

WCopula

The lower Fréchet–Hoeffding bound is the copula with the smallest value among all copulas. Note that $W$ is a proper copula only when $d=2$; for $d>2$ it remains the pointwise lower bound but is not itself a copula. For any copula $C$ and all $\mathbf{u}\in [0,1{]}^d$,

$$W(\mathbf{u})\le C(\mathbf{u})\le M(\mathbf{u}).$$

Both Fréchet–Hoeffding bounds are Archimedean copulas.

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

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Transformed Copulas

SurvivalCopula

Copulas.SurvivalCopula Type

SurvivalCopula(C, flips)
SurvivalCopula{d,CT}

Construct the survival (flipped) version of a copula by flipping the arguments at the given indices.

The ergonomic constructor SurvivalCopula(C, flips) accepts the indices to flip:

SurvivalCopula(ClaytonCopula(4, θ), (2,3))

The flip pattern is stored in the object rather than its type, so distinct rotations of the same copula share one concrete type.

For a copula C in dimension d and indices i₁, ..., iₖ ∈ 1:d, the survival copula flips the corresponding arguments:

$$S(u_1,\dots ,u_d)=C(v_1,\dots ,v_d),v_j=\{\begin{array}{ll}1-u_j& j\in \text{flips}\\ u_j& \text{otherwise}\end{array}$$

Notes:

• In the bivariate case, this includes the usual 90/180/270-degree "rotations" of a copula family.

• The resulting object is handled like the base copula: same API (cdf, pdf/logpdf, rand, fit) and uniform marginals in $[0,1{]}^d$.

References:

• [3] Nelsen (2006), An introduction to copulas.

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Others

PlackettCopula

Copulas.PlackettCopula Type

PlackettCopula{d, P}

Fields: - θ::Real - parameter

Constructor

PlackettCopula(θ)

The Plackett copula is parameterized by $\theta >0$ and is defined by

$$C_{\theta }(u,v)=\frac{[1+(\theta -1)(u+v)]-\sqrt{[1+(\theta -1)(u+v){]}^2-4uv\theta (\theta -1)}}{2(\theta -1)}$$

and for $\theta =1$ we have $C_1(u,v)=uv$.

Special cases:

• θ = 0: WCopula (lower Fréchet–Hoeffding bound)

• θ = 1: IndependentCopula

• θ = ∞: MCopula (upper Fréchet–Hoeffding bound)

These values retain the concrete PlackettCopula family type.

References:

• [4] Joe, H. (2014). Dependence modeling with copulas. CRC press, Page.164

• [75] Johnson, Mark E. Multivariate statistical simulation: A guide to selecting and generating continuous multivariate distributions. Vol. 192. John Wiley & Sons, 1987. Page 193.

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006. Exercise 3.38.

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FGMCopula

Farlie-Gumbel-Morgenstern (FGM) copula

Copulas.FGMCopula Type

FGMCopula{d,T}

Fields:

• θ::Real - parameter

Constructor

FGMCopula(d, θ)

The multivariate Farlie–Gumbel–Morgenstern (FGM) copula of dimension d has $2^d-d-1$ parameters $\theta$ and

$$C(\mathit{u})=\prod _{i=1}^d{u}_i[1+\sum _{k=2}^d\sum _{1\le j_1<\cdots <j_k\le d}{\theta }_{j_1\cdots j_k}{\overline{u}}_{j_1}\cdots {\overline{u}}_{j_k}],$$

where $\overline{u}=1-u$.

Special cases:

• When d=2 and θ = 0, it is the IndependentCopula.

More details about Farlie-Gumbel-Morgenstern (FGM) copula are found in [3]. We use the stochastic representation from [76] to obtain random samples.

References:

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.

• [76] Blier-Wong, C., Cossette, H., & Marceau, E. (2022). Stochastic representation of FGM copulas using multivariate Bernoulli random variables. Computational Statistics & Data Analysis, 173, 107506.

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RafteryCopula

Copulas.RafteryCopula Type

RafteryCopula{d, P}

Fields: - θ::Real - parameter

Constructor

RafteryCopula(d, θ)

The multivariate Raftery copula of dimension d is parameterized by $\theta \in [0,1]$.

$$C_{\theta }(\mathbf{u})=u_{(1)}+\frac{(1-\theta )(1-d)}{1-\theta -d}{\left(\prod _{j=1}^d{u}_j\right)}^{\frac{1}{1-\theta }}-\sum _{i=2}^d\frac{\theta (1-\theta )}{(1-\theta -i)(2-\theta -i)}{\left(\prod _{j=1}^{i-1}{u}_{(j)}\right)}^{\frac{1}{1-\theta }}{u}_{(i)}^{\frac{2-\theta -i}{1-\theta }}$$

where $u_{(1)},\dots ,u_{(d)}$ denote the order statistics of $u_1,\dots ,u_d$. More details about Multivariate Raftery Copula are found in the references below.

Special cases:

• When θ = 0, it is the IndependentCopula.

• When θ = 1, it is the the Fréchet upper bound

References:

• [77] Saali, T., M. Mesfioui, and A. Shabri, 2023: Multivariate Extension of Raftery Copula. Mathematics, 11, 414, https://doi.org/10.3390/math11020414.

• [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006. Exercise 3.6.

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References

2. R. B. Nelsen. An Introduction to Copulas. 2nd ed Edition, Springer Series in Statistics (Springer, New York, 2006).

3. H. Joe. Dependence Modeling with Copulas (CRC press, 2014).

4. M. E. Johnson. Multivariate statistical simulation: A guide to selecting and generating continuous multivariate distributions. Vol. 192 (John Wiley & Sons, 1987).

5. C. Blier-Wong, H. Cossette and E. Marceau. Stochastic representation of FGM copulas using multivariate Bernoulli random variables. Computational Statistics & Data Analysis 173, 107506 (2022).

6. T. Saali, M. Mesfioui and A. Shabri. Multivariate extension of Raftery copula. Mathematics 11, 414 (2023).