The t copula

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The t copula is the copula that underlies the multivariate Student’s t distribution.


Copula name

t copula

Common notation


, a non-negative definite  matrix, i.e. a matrix that can correspond to a correlation matrix

 = degrees of freedom (, usually  is an integer although in some situations a non-integral  can arise)

(note in principle each marginal distribution could in principle have a different number of degrees of freedom although such a refinement is not commonly seen)



where  is the inverse student’s t function and  is the cumulative distribution function of the multivariate student’s t distribution with arbitrary mean and matrix generator equal to

Kendall’s rank correlation coefficient (for bivariate case),

Where  is the correlation coefficient between the two variables

Coefficient of upper tail dependence,

Coefficient of lower tail dependence,

Other comments

If  (the  identity matrix) then, in contrast to the Gaussian copula, we do not recover the independence copula.


Nematrian web functions


Functions relating to the above distribution in the two-dimensional case may be accessed via the Nematrian web function library by using a DistributionName of “student’s t (2d)”. For details of other supported probability distributions see here.


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