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How to calculate the log likelihood of an Archimedean copula?
The numerical algorithm for root finding can be used to invert the function to obtain as a function of . Alternatively, you can use the MLE or the CMLE method to estimate the parameter given the data and . The log-likelihood function for each type of Archimedean copula is provided in the following sections.
Which is the generator of a Clayton copula?
A Clayton copula is defined as with . with for and for . Let the generator function . A Gumbel copula is defined as with . Suppose the generator of the Archimedean copula is . Then the simulation method using Laplace-Stieltjes transformation of the distribution function is given by Marshall and Olkin (1988) where :
Is the Frank copula related to a logarithmic random variable?
For the Frank copula with , , and is a discrete probability function . This probability function is related to a logarithmic random variable with parameter value . For details about simulating a random variable from a stable distribution, see Theorem 1.19 in Nolan (2010).
Is the Gumbel copula a discrete probability function?
For the Gumbel copula, , and is the distribution function of the stable variable with . For the Frank copula with , , and is a discrete probability function .
What kind of Tau does an Archimedean copula have?
A bivariate Archimdean copula has a Kendall’s tau,, as follows: Where the copula depends on just one parameter then this allows the parameter to be estimated from observed data.
Which is the generator function of the Archimedean copula?
Many of the copulas tabulated above are Archimedean copulas. These are copulas which take the form: Here, is called the generator function of the Archimedean copula and is its inverse function.
How are bivariate Archimedean copulas used in modelling?
Bivariate Archimedean copulas are often used in modelling applications. However, as the number of dimensions increases it becomes increasingly implausible that the ‘true’ dependency structure should be exchangeable across all possible variables simultaneously.