param = copulaparam (family, r) returns the linear or copula parameter param corresponding to a copula of the family family that has Kendall’s rank correlation r. It is the inverse of copulastat.

family is the copula family name. It can be "Gaussian" for the Gaussian family, "t" for the Student’s t family, "Clayton" for the Clayton family, "Gumbel" for the Gumbel-Hougaard family, "Frank" for the Frank family, "AMH" for the Ali-Mikhail-Haq family, or "FGM" for the Farlie-Gumbel-Morgenstern family. The last two are Octave extensions that MATLAB does not provide, and are treated as bivariate. Neither reaches the whole range of either rank correlation: the Ali-Mikhail-Haq family covers a Kendall’s tau in [(5-8*log (2))/3, 1/3] and the Farlie-Gumbel-Morgenstern family one in [-2/9, 2/9].

For the Gaussian and Student’s t families, r is a scalar rank correlation or a matrix of pairwise rank correlations, and param is the corresponding linear correlation of the same size. For the Clayton, Gumbel-Hougaard, and Frank families, r is a scalar rank correlation and param is the scalar copula parameter. The Gumbel-Hougaard family models positive dependence only, so r must be non-negative for that family.

param = copulaparam (…, "type", type) selects the measure of rank correlation given in r. type can be "Kendall" (the default) for Kendall’s tau, or "Spearman" for Spearman’s rho.

The Gaussian and Student’s t families and the Kendall’s tau of the Clayton and Gumbel-Hougaard families are inverted in closed form. The remaining cases are inverted numerically from copulastat.

Note: for the Archimedean families with "Spearman", the underlying relationship is computed by exact numerical integration rather than the interpolated table used by MATLAB, so results may differ from MATLAB by up to about $10^\{-4\}$. See copulastat.

See also: copulastat, copulafit, copulacdf, copulapdf, copularnd

Source Code: copulaparam