This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

 CopulaDistribution represents a copula distribution with kernel distribution ker and marginal distributions , , ....
• The cumulative distribution function is given by , where is the CDF for the kernel ker, and is the CDF for .
• The following kernels ker can be used:
 "Product" independent distributions "Maximal" Frechét-Hoeffding upper bound "Minimal" Frechét-Hoeffding lower bound {"Frank",} Frank copula {"Clayton",c} Clayton-Pareto copula {"GumbelHougaard",} Gumbel-Hougaard copula {"FGM",} Farlie-Gumbel-Morgenstern copula {"AMH",} Ali-Mikhail-Haq copula {"Binormal",} bivariate Gaussian with correlation {"Multinormal",} multivariate Gaussian with covariance {"MultivariateT",,} multivariate -distribution with scale matrix and degrees of freedom
• For , can be any positive number.
• For , can be any positive number.
• For , can be any real number greater than or equal to 1.
• For and , can be any real number between and .
Define a product copula:
Define a Farlie-Gumbel-Morgenstern copula:
Define a three-dimensional maximal copula:
Define a product copula:
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Define a Farlie-Gumbel-Morgenstern copula:
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Define a three-dimensional maximal copula:
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 Scope   (31)
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:
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 Gumbel-Hougaard copula:
Probability density function:
Cumulative distribution function:
A Farlie-Gordon-Morgenstern copula:
Probability density function:
Cumulative distribution function:
An Ali-Mikhail-Haque 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:
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:
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:
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:
 Applications   (5)
A system is composed of four components, each with lifespan exponentially distributed with parameter . Dependencies in the time to failure are modeled by Farlie-Gumbel-Morgenstern copula with . 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 :
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 Gumbel-Hougaard copula for different values of the parameter:
Show how the value of the parameter influences the dependence between values:
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 normal marginals is a MultivariateTDistribution:
MarginalDistribution of a copula returns the component distributions:
Several copula kernels with uniform marginals:
A Frank copula with different marginals:
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