Gumbel Copula Functions

Functions
real	gumbel_copula_lpdf (real u, real v, real theta)

Detailed Description

 
real gumbel_copula_lpdf(real u, real v, real theta) {
  real neg_log_u = -log(u);
  real log_neg_log_u = log(neg_log_u);
  real neg_log_v = -log(v);
  real log_neg_log_v = log(neg_log_v);
  real log_temp = log_sum_exp(theta * log_neg_log_u, theta * log_neg_log_v);
  real theta_m1 = theta - 1;
  
  if (theta < 1) 
    reject("theta must be >= 1");
  
  if (is_inf(theta)) {
    if (u == v) 
      return 0;
    else 
      return negative_infinity();
  }
  
  return theta_m1 * log_neg_log_u + theta_m1 * log_neg_log_v + neg_log_u + neg_log_v
         - exp(log_temp / theta)
         + log_sum_exp(2 * theta_m1 / -theta * log_temp,
                       log(theta_m1) + (1 - 2 * theta) / theta * log_temp);
}
 

Function Documentation

gumbel_copula_lpdf()

real gumbel_copula_lpdf	(	real	u,
		real	v,
		real	theta
	)

Gumbel Copula Log Density

Archimedean family of Gumbel with parametric generator

\[ \exp(-t^{1/\theta}), t \in [0, \infty] \]

with \(\theta \in [1, \infty)\). The range of admissible Kendall’s tau, as well as that of the upper taildependence coefficient, is \([0,1)\). Note that this copula does not allow for lower tail dependence.

https://cran.r-project.org/web/packages/copula/copula.pdf "copula" R package in the copFamilies documentation accessed Feb. 5, 2021

Copyright Ben Goodrich https://groups.google.com/g/stan-users/c/hnUtkMYlLhQ/m/UZURBv2_AAAJ Accessed Feb. 5, 2021

Parameters

u	Real number on (0,1], not checked but function will return NaN
v	Real number on (0,1], not checked but function will return NaN
theta	Real number >= 1, will throw otherwise
log	density