W_N5p12a {copBasic}

Ordinal Sums of Lower-Bound Copula, Example 5.12a of Nelsen's Book

Description

Compute shuffles of Fréchet–Hoeffding lower-bound copula (Nelsen, 2006, p. 173), which is defined by partitioning \mathbf{W} within \mathcal{I}^2 into n subintervals:

\mathbf{W}_n(u,v) = \mathrm{max}\biggl(\frac{k-1}{n}, u+v-\frac{k}{n} \biggr)

for points within the partitions

(u,v) \in \biggl[\frac{k-1}{n}, \frac{k}{n}\biggr]\times \biggl[ \frac{k-1}{n}, \frac{k}{n}\biggr]\mbox{,\ }k = 1,2,\cdots,n

and for points otherwise out side the partitions

\mathbf{W}_n(u,v) = \mathrm{min}(u,v)\mbox{.}

The support of \mathbf{W}_n consists of n line segments connecting coordinate pairs \{(k-1)/n,\, k/n\} and \{k/n,\, (k-1)/n\} as stated by Nelsen (2006). The Spearman Rho (rhoCOP) is defined by \rho_\mathbf{C} = 1 - (2/n^2), and the Kendall Tau (tauCOP) by \tau_\mathbf{C} = 1 - (2/n).

Usage

W_N5p12a(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

W, ORDSUMcop, ORDSUWcop, M_N5p12b

Examples

W_N5p12a(0.4, 0.6, para=5)

## Not run: 
  # Nelsen (2006, exer. 5.12a, p. 172, fig. 5.3a)
  UV <- simCOP(1000, cop=W_N5p12a, para=4) # which is the same as
  para <- list(cop=c(W, W, W, W), para=NULL, part=c(0,0.25,0.50,0.75,1))
  UV <- simCOP(1000, cop=ORDSUMcop, para=para) 
## End(Not run)