represents a copula distribution with kernel distribution ker and marginal distributions dist1, dist2, .

# Details

• 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" Frechét–Hoeffding upper bound "Minimal" Frechét–Hoeffding lower bound {"Frank",α} Frank copula {"Clayton",c} Clayton–Pareto copula {"GumbelHougaard",α} Gumbel–Hougaard copula {"FGM",α} Farlie–Gumbel–Morgenstern copula {"AMH",α} Ali–Mikhail–Haq 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 a certain constant for dimensions higher than two.
• 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

• CopulaDistribution[ker,{dist1,dist2,,distn}] represents a multivariate statistical distribution whose 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 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.

# Examples

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## Basic Examples(3)

Define a product copula:

Define a FarlieGumbelMorgenstern copula:

Define a three-dimensional maximal copula:

## Scope(32)

### Basic Uses(6)

Define a product copula using two normal distributions:

Probability density function:

Cumulative distribution function:

Define a Frank copula using two uniform distributions:

Generate random vectors:

Compare means and variances:

Define an FGM copula with beta distributions:

Moment and moment-generating function:

Define a maximal copula with discrete components:

Probability density function:

Compute probabilities and expectations:

Define a minimal copula with Poisson distributions:

Probability density function:

Statistical properties are calculated componentwise:

Estimate distribution parameters:

### Copula Kernels(11)

A product copula:

Probability density function:

Cumulative distribution function:

A maximal copula:

Cumulative distribution function:

A minimal copula:

Cumulative distribution function:

A Frank copula:

Probability density function:

Cumulative distribution function:

A Clayton copula:

Probability density function:

Cumulative distribution function:

A GumbelHougaard copula:

Probability density function:

Cumulative distribution function:

A FarlieGordonMorgenstern copula:

Probability density function:

Cumulative distribution function:

An AliMikhailHaque copula:

Probability density function:

Cumulative distribution function:

A binormal copula:

Probability density function:

A multinormal copula:

Probability density function:

A multivariate Student copula:

Probability density function:

### Parametric Distributions(4)

Define a minimal copula with beta distributions as marginals:

Cumulative distribution function:

Survival function:

Define a maximal copula with different continuous marginals:

Cumulative distribution function:

Mean and variance:

Skewness and kurtosis:

Define a copula with Poisson marginal distributions:

Probability density function:

Hazard function:

Define a copula with negative binomial distribution marginals:

Probability density function:

Generate random numbers:

### Nonparametric Distributions(3)

Define a copula with SmoothKernelDistribution:

Probability density function:

Mean and variance:

Define a copula with EmpiricalDistribution:

Probability density function:

Define a copula with a HistogramDistribution:

Plot probability density function:

Cumulative distribution function:

### Derived Distributions(8)

Define a copula distribution with a TruncatedDistribution as a marginal:

Probability density function:

Define a copula distribution with a CensoredDistribution as a marginal:

Probability density function:

Mean and variance:

Define a copula with a MixtureDistribution as a marginal:

Probability density function:

Cumulative distribution function:

Define a copula with a ParameterMixtureDistribution as a marginal:

Probability density function:

Hazard function:

Define a copula with an OrderDistribution as a marginal:

Probability density function:

Cumulative distribution function:

Define a copula with a TransformedDistribution as a marginal:

Probability density function:

Mean and variance:

Skewness and kurtosis:

Define a copula with a MarginalDistribution as a marginal:

Probability density function:

Copula with QuantityDistribution marginals evaluates to QuantityDistribution:

Mean and variance:

## Applications(6)

A system is composed of four components, each with lifespan exponentially distributed with parameter per hour. Dependencies in the time to failure are modeled by a FarlieGumbelMorgenstern copula with α1/3. Find the probability that no component fails before 500 hours:

Find the probability that one component will fail after 1000 hours:

Assume the values of two assets follow a geometric Brownian motion with drifts and and volatilities and , respectively. Assuming both initial values to be 1, find the bounds for the joint cumulative distribution function of both assets at time :

Lower bound:

Upper bound:

Assuming the values below compare the plots of the CDFs:

Two firms have debts and and initial assets both equal to 1. Assume the values of the assets follow a geometric Brownian motion with drifts and and volatilities and , respectively. Find the joint probability of the default at time assuming a Frank copula:

Default probability depending on α:

Limiting values:

A Cauchy copula is a multivariate Student copula with one degree of freedom:

Probability density function:

Visualize the density using a scatter plot:

Define a GumbelHougaard copula for different values of the parameter:

Show how the value of the parameter influences the dependence between values:

Gumbel's bivariate logistic distribution is an AMH copula with logistic marginal distributions:

Visualize its probability density function:

Cumulative distribution function has the structure of CDF of the univariate logistic distribution:

## Properties & Relations(5)

The product copula distribution of two normal distributions is a binormal distribution:

Product copula is equivalent to binormal copula with zero correlation:

Binormal copula with normal marginals is a BinormalDistribution:

Multivariate copula with Student marginals is a MultivariateTDistribution:

MarginalDistribution of a copula returns the component distributions:

## Possible Issues(1)

CopulaDistribution will not accept ProductDistribution as a marginal:

The correct syntax is to enter all the distributions in the list:

## Neat Examples(2)

Several copula kernels with uniform marginals:

A Frank copula with different marginals: