Compute shuffles of Fréchet--Hoeffding upper-bound copula (Nelsen, 2006, p. 173), which is defined by partitioning \(\mathbf{M}\) within \(\mathcal{I}^2\) into \(n\) subintervals:
$$\mathbf{M}_n(u,v) = \mathrm{min}\biggl(u-\frac{k-1}{n}, v-\frac{n-k}{n} \biggr)$$
for points within the partitions
$$(u,v) \in \biggl[\frac{k-1}{n}, \frac{k}{n}\biggr]\times \biggl[ \frac{n-k}{n}, \frac{n-k+1}{n}\biggr]\mbox{,\ }k = 1,2,\cdots,n$$
and for points otherwise out side the partitions
$$\mathbf{M}_n(u,v) = \mathrm{max}(u+v-1,0)\mbox{.}$$
The support of \(\mathbf{M}_n\) consists of \(n\) line segments connecting coordinate pairs \(\{(k-1)/n,\, (n-k)/n\}\) and \(\{k/n,\, (n-k+1)/n\}\) as stated by Nelsen (2006). It is useful that Nelsen stated such as this helps to identify that Nelsen's typesetting of the terms in the second square brackets---the \(V\) direction---is reversed from that shown in this documentation. The Spearman Rho (rhoCOP) is defined by \(\rho_\mathbf{C} = (2/n^2) - 1\), and the Kendall Tau (tauCOP) by \(\tau_\mathbf{C} = (2/n) - 1\).
M_N5p12b(u, v, para=1, ...)Value(s) for the copula are returned.
Nonexceedance probability \(u\) in the \(X\) direction;
Nonexceedance probability \(v\) in the \(Y\) direction;
A positive integer \(n \in 1, 2, \cdots\); and
Additional arguments to pass.
W.H. Asquith
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
M, ORDSUMcop, W_N5p12a