Multivariate Student_t_copula_cholesky_lpdf assumption

Modeling
1

The following outlines an implementation of the multivariate Gaussian copula:

  real gauss_copula_cholesky_lpdf(matrix u, matrix L) {
    array[rows(u)] row_vector[cols(u)] q;
    for (n in 1:rows(u)) {
      q[n] = std_normal_log_qf(u[n]);
    }
    return multi_normal_cholesky_lpdf(q | rep_row_vector(0, cols(L)), L)
             - std_normal_lpdf(to_vector(to_matrix(q)));
  }

Consequently, it is presumed that the multivariate Student-t copula is:

  real studentt_copula_cholesky_lpdf(matrix u, real nu, matrix L) {
    real sd_nu = 1.0; //inv_sqrt(nu /( nu - 2.0)); // this was an idea to standardise the variance to 1.
    array[rows(u)] row_vector[cols(u)] q;
    for (n in 1:rows(u))  
      q[n] = student_t_log_qf(u[n], nu);

    return multi_student_t_cholesky_lpdf(q | nu, rep_row_vector(0, cols(L)), diag_pre_multiply(rep_vector(sd_nu, cols(L)), L))
            - student_t_lpdf(to_vector(u) | nu, 0, sd_nu);
  }

Could an expert kindly confirm this?

2

I haven’t done the math but it looks like others do the same here. See It was the best of tails, it was the worst of tails: The T-Copula – bggj.

1 Like
3

Yep it’s the same for all multivariate densities, as :

\begin{align} c(u_1,\ldots,u_n;\Sigma)&=\frac{\partial^nC(u_1,.\ldots,u_n;\Sigma)}{\partial u_1\cdots\partial u_n}=\frac{f(x;0,\Sigma)}{\prod_{i=1}^n f(x_i)} \end{align}
1 Like