CopulaDistribution

CopulaDistribution[ker,{dist1,dist2,}]
represents a copula distribution with kernel distribution ker and marginal distributions dist1, dist2, .

DetailsDetails

  • The cumulative distribution function is given by , where is the CDF for the kernel ker, and is the CDF for disti.
  • Marginal distribution disti can be any univariate distribution.
  • The following kernels ker can be used:
  • "Product"independent distributions
    "Maximal"FrechétHoeffding upper bound
    "Minimal"FrechétHoeffding lower bound
    {"Frank",α}Frank copula
    {"Clayton",c}ClaytonPareto copula
    {"GumbelHougaard",α}GumbelHougaard copula
    {"FGM",α}FarlieGumbelMorgenstern copula
    {"AMH",α}AliMikhailHaq copula
    {"Binormal",ρ}bivariate Gaussian with correlation
    {"Multinormal",Σ}multivariate Gaussian with covariance
    {"MultivariateT",Σ,ν}multivariate -distribution with scale matrix and degrees of freedom
  • For "Frank", can be any positive number in two dimensions and any positive number less than or equal to 1 in higher dimensions.
  • For "Clayton", can be any positive number.
  • For "GumbelHougaard", can be any real number greater than or equal to 1.
  • For "FGM" and "AMH", can be any real number between and .
  • The parameters for "Binormal", "Multinormal", and "MultivariateT" are the same as for BinormalDistribution, MultinormalDistribution, and MultivariateTDistribution, respectively.
  • CopulaDistribution can be used with such functions as Mean, PDF, and RandomVariate, etc.

Background & Context
Background & Context

  • CopulaDistribution[ker,{dist1,dist2,,distn}] represents a multivariate statistical distribution whose ^(th)marginal distribution (MarginalDistribution) is precisely distj, and for which the CDF of a distj-distributed random variate follows a uniform distribution (UniformDistribution). For a general copula distribution CopulaDistribution[ker,{dist1,dist2,,distn}], the probability density function (PDF) of Yj=TransformedDistribution[Fj[x],xdistj] is equivalent to UniformDistribution[] whenever Fj[x] is the CDF of distj. While all copula distributions share the above properties, the characteristics and behavior of a specific copula distribution depend both on its kernel ker and on its marginals dist1,dist2,,distn.
  • In practice, a copula is a tool that describes dependence between variables, and in this context, varying ker allows investigation of different degrees of dependence (for example, {"FGM",α} best models weak variable dependence, whereas "Product" allows analysis of independent variables). There are 11 predefined kernels ker that may be used to parametrize a copula distribution. These 11 can be split into roughly four groups, consisting of the independence-dependence kernels ("Product", "Maximal", and "Minimal"); the Archimedean kernels ({"Frank",α}, where for and for , {"Clayton",c} for , {"GumbelHougaard",α} for , and {"AMH",α} for ); the distribution-derived kernels ({"Binormal",ρ} for ρ as in BinormalDistribution, {"Multinormal",Σ} for Σ as in MultinormalDistribution, and {"MultivariateT",Σ,ν} for Σ, ν as in MultivariateTDistribution); and the non-associative kernels ({"FGM",α} for ), members of which share similar qualitative or theoretical properties.
  • Sklar's theorem proves the existence of a copula that "couples" any joint distribution with its univariate marginals via the relation and thus demonstrates that copula distributions are ubiquitous in multivariate statistics. Copula distributions date as far back as the 1940s, though much of the terminology and machinery used today were developed in the 1950s and 1960s. Since their inception, copulas have been used to model phenomena in areas including reliability theory, meteorology, and queueing theory, while specially purposed copulas and kernels have been developed to serve as tools in fields such as survival analysis (via survival copulas) and mathematical finance (via panic copulas). Copula distributions are also of independent theoretical interest in Monte Carlo theory and applied mathematics.
  • Many relationships exist between CopulaDistribution[ker,{dist1,,distn}] and various other distributions depending on the parameters ker and distj. CopulaDistribution["Product",{dist1,,distn}] is equivalent to ProductDistribution[dist1,,distn] for all distributions distj, and so the product copula of two instances of NormalDistribution is BinormalDistribution. In addition, the product copula is equivalent to the binormal copula with zero correlation in the sense that the PDF of CopulaDistribution["Product",{dist1,,distn}] is precisely the same as that of CopulaDistribution[{"Binormal",0},{dist1,,distn}] for all distributions distj. Among distribution-derived kernels, a binormal copula with NormalDistribution marginals and a multivariate -copula with StudentTDistribution marginals are equivalent to BinormalDistribution and MultivariateTDistribution, respectively, while a practically limitless number of qualitatively similar relationships exist between Archimedean copulas and miscellaneous distributions.

ExamplesExamplesopen allclose all

Basic Examples  (3)Basic Examples  (3)

Define a product copula:

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Define a FarlieGumbelMorgenstern copula:

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Define a three-dimensional maximal copula:

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Introduced in 2010
(8.0)
| Updated in 2016
(10.4)