Other Copulas
A few copulas, while necessary in certain cases and really useful, are hard to classify. We gather them here for simplicity.
PlackettCopula
Copulas.PlackettCopula
— TypePlackettCopula{P}
Fields: - θ::Real - parameter
Constructor
PlackettCopula(θ)
Parameterized by $\theta > 0$ The Plackett copula is
\[C_{\theta}(u,v) = \frac{\left [1+(\theta-1)(u+v)\right]- \sqrt{[1+(\theta-1)(u+v)]^2-4uv\theta(\theta-1)}}{2(\theta-1)}\]
and for $\theta = 1$
\[C_{1}(u,v) = uv \]
It has a few special cases:
- When θ = 0, is is the MCopula (Upper Frechet-Hoeffding bound)
- When θ = 1, it is the IndependentCopula
- When θ = ∞, is is the WCopula (Lower Frechet-Hoeffding bound)
References:
- [4] Joe, H. (2014). Dependence modeling with copulas. CRC press, Page.164
- [56] 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.
FGMCopula
Farlie-Gumbel-Morgenstern (FGM) copula
Copulas.FGMCopula
— TypeFGMCopula{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 function
\[C(\boldsymbol{u})=\prod_{i=1}^{d}u_i \left[1+ \sum_{k=2}^{d}\sum_{1 \leq j_1 < \cdots < j_k \leq d} \theta_{j_1 \cdots j_k} \bar{u}_{j_1}\cdots \bar{u}_{j_k} \right],\]
where $\bar{u}=1-u$.
More details about Farlie-Gumbel-Morgenstern (FGM) copula are found in :
Nelsen, Roger B. An introduction to copulas. Springer, 2006. Exercise 3.38.
We use the stochastic representation of the copula to obtain random samples.
Blier-Wong, C., Cossette, H., & Marceau, E. (2022). Stochastic representation of FGM copulas using multivariate Bernoulli random variables. Computational Statistics & Data Analysis, 173, 107506.
It has a few special cases:
- When d=2 and θ = 0, it is the IndependentCopula.
References:
- [3] Nelsen, Roger B. An introduction to copulas. Springer, 2006.
RafteryCopula
Copulas.RafteryCopula
— TypeRafteryCopula{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 $u_{(1)}, \ldots , u_{(d)}$ denote the order statistics of $u_1, \ldots ,u_d$. More details about Multivariate Raftery Copula are found in the references below.
It has a few special cases:
- When θ = 0, it is the IndependentCopula.
- When θ = 1, it is the the Fréchet upper bound
References:
- [15]
- M.-P. Côté and C. Genest. Dependence in a Background Risk Model. Journal of Multivariate Analysis 172, 28–46 (2019).
- [24]
- M. Hofert. Sampling Nested Archimedean Copulas with Applications to CDO Pricing. Ph.D. Thesis, Universität Ulm (2010).
- [25]
- M. Hofert and D. Pham. Densities of Nested Archimedean Copulas. Journal of Multivariate Analysis 118, 37–52 (2013).
- [26]
- A. J. McNeil and J. Nešlehová. From Archimedean to Liouville Copulas. Journal of Multivariate Analysis 101, 1772–1790 (2010).
- [27]
- H. Cossette, S.-P. Gadoury, E. Marceau and I. Mtalai. Hierarchical Archimedean Copulas through Multivariate Compound Distributions. Insurance: Mathematics and Economics 76, 1–13 (2017).
- [28]
- H. Cossette, E. Marceau, I. Mtalai and D. Veilleux. Dependent Risk Models with Archimedean Copulas: A Computational Strategy Based on Common Mixtures and Applications. Insurance: Mathematics and Economics 78, 53–71 (2018).
- [29]
- C. Genest, J. Nešlehová and J. Ziegel. Inference in Multivariate Archimedean Copula Models. TEST 20, 223–256 (2011).
- [30]
- E. Di Bernardino and D. Rulliere. On Certain Transformations of Archimedean Copulas: Application to the Non-Parametric Estimation of Their Generators. Dependence Modeling 1, 1–36 (2013).
- [31]
- E. Di Bernardino and D. Rullière. On an Asymmetric Extension of Multivariate Archimedean Copulas Based on Quadratic Form. Dependence Modeling 4 (2016).
- [32]
- K. Cooray. Strictly Archimedean Copulas with Complete Association for Multivariate Dependence Based on the Clayton Family. Dependence Modeling 6, 1–18 (2018).
- [33]
- J. Spreeuw. Archimedean Copulas Derived from Utility Functions. Insurance: Mathematics and Economics 59, 235–242 (2014).