copBasic (version 2.2.6)

W_N5p12a: 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, ...)

Value

Value(s) for the copula are returned.

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.

Author

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)

if (FALSE) {
  # 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) }