Compute the lower semi-correlations (bottom-left)
$$\rho^{N{-}{-}}_\mathbf{C}(u,v; a) = \rho_N^{{-}{-}}(a) \mbox{ and}$$
compute the upper semi-correlations (top-right)
$$\rho^{N{+}{+}}_\mathbf{C}(u,v; a) = \rho_N^{{+}{+}}(a)$$
of a copula \(\mathbf{C}(u,v)\) (Joe, 2014, p. 73) using numerical simulation. The semi-correlations are defined as
$$\rho_N^{{-}{-}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 < -a, Z_2 < -a]\mbox{,}$$
$$\rho_N^{{+}{+}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 > +a, Z_2 > +a]\mbox{,\ and}$$
$$\rho_N(a > -\infty) = \mathrm{cor}[Z_1, Z_2]\mbox{,}$$
where \(\mathrm{cor[z_1, z_2]}\) is the familiar Pearson correlation function, which is in R the syntax cor(..., method="pearson"), parameter \(a \ge 0\) is a truncation point that identifies truncated tail regions (Joe, 2014, p. 73), and lastly \((Z_1, Z_2) \sim \mathbf{C}(\Phi, \Phi)\) and thus from the standard normal distribution \((Z_1, Z_2) = (\Phi^{-1}(u), \Phi^{-1}(v))\) where the random variables \((U,V) \sim \mathbf{C}\).
The semi-correlations are extended for the copBasic package into bottom right and top left versions as well by $$\rho_N^{{+}{-}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 > +a, Z_2 < -a]\mbox{,\ and}$$ $$\rho_N^{{-}{+}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 < -a, Z_2 > +a]\mbox{.}$$ As a result, the notations \({-}{-}\), \({+}{+}\), \({+}{-}\), and \({-}{+}\) can be used to represent each of the respective corners bottom-left, top-right, bottom-right, and top-left of the \((u,v)\) domain with the respective truncation. These words are used in the variable names of the returned list from the function.
semicorCOP(cop=NULL, para=NULL, truncation=0, n=0, as.sample=FALSE, ...)The value(s) for \(\rho_N\), \(\rho_N^{{-}{-}}\), \(\rho_N^{{+}{+}}\), \(\rho_N^{{+}{-}}\), and \(\rho_N^{{-}{+}}\) are returned.
A copula function;
Vector of parameters or other data structure, if needed, to pass to the copula;
The truncation value for \(a\), which is in standard normal variates, and the default of zero is the origin (medians);
The sample size \(n\) for simulation estimates of the \(\rho_N\);
A logical controlling whether an optional data.frame in para is used to compute the \(\hat\rho_N\) (see Note); and
Additional arguments to pass to the copula.
W.H. Asquith
Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.
giniCOP, rhoCOP, tauCOP, COP