HPCOPULA Procedure

Student’s t Copula

Let normal upper Theta equals left-brace left-parenthesis nu comma normal upper Sigma right-parenthesis colon nu element-of left-parenthesis 1 comma normal infinity right-parenthesis comma normal upper Sigma element-of double-struck upper R Superscript m times m Baseline right-brace, and let t Subscript nu be a univariate t distribution with nu degrees of freedom.

The Student’s t copula can be written as

upper C Subscript normal upper Theta Baseline left-parenthesis u 1 comma u 2 comma ellipsis comma u Subscript m Baseline right-parenthesis equals t t Subscript nu comma normal upper Sigma Baseline left-parenthesis t Subscript nu Superscript negative 1 Baseline left-parenthesis u 1 right-parenthesis comma t Subscript nu Superscript negative 1 Baseline left-parenthesis u 2 right-parenthesis comma ellipsis comma t Subscript nu Superscript negative 1 Baseline left-parenthesis u Subscript m Baseline right-parenthesis right-parenthesis

where t t Subscript nu comma normal upper Sigma is the multivariate Student’s t distribution that has a correlation matrix normal upper Sigma with nu degrees of freedom.

Simulation

The input parameters for the simulation are left-parenthesis nu comma normal upper Sigma right-parenthesis. The t copula can be simulated by the following steps:

  1. Generate a multivariate vector bold-italic upper X tilde t Subscript m Baseline left-parenthesis nu comma 0 comma normal upper Sigma right-parenthesis that follows the centered t distribution with nu degrees of freedom and correlation matrix normal upper Sigma.

  2. Transform the vector bold-italic upper X into bold-italic upper U equals left-parenthesis t Subscript nu Baseline left-parenthesis upper X 1 right-parenthesis comma ellipsis comma t Subscript nu Baseline left-parenthesis upper X Subscript m Baseline right-parenthesis right-parenthesis Superscript upper T, where t Subscript nu is the distribution function of univariate t distribution with nu degrees of freedom.

To simulate centered multivariate t random variables, you can use the property that bold-italic upper X tilde t Subscript m Baseline left-parenthesis nu comma 0 comma normal upper Sigma right-parenthesis if bold-italic upper X equals StartRoot nu slash s EndRoot bold-italic upper Z, where bold-italic upper Z tilde upper N left-parenthesis 0 comma normal upper Sigma right-parenthesis and the univariate random variable s tilde chi Subscript nu Superscript 2.

Last updated: June 19, 2025