M_N5p12b {copBasic}

Shuffles of Upper-Bound Copula, Example 5.12b of Nelsen's Book

Description

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.

Usage

M_N5p12b(u, v, para=1, ...)

Arguments

u

Nonexceedance probability u in the X direction;

v

Nonexceedance probability v in the Y direction;

para

A positive integer n \in 1, 2, \cdots; and

...

Additional arguments to pass.

Value

Value(s) for the copula are returned.

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

See Also

M, ORDSUMcop, W_N5p12a

Examples

M_N5p12b(0.4, 0.6, para=3)

## Not run: 
  # Nelsen (2006, exer. 5.12b, p. 173, fig. 5.3b)
  UV <- simCOP(1000, cop=M_N5p12b, para=4) #
## End(Not run)