W_N5p12a {copBasic}R Documentation

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 W\mathbf{W} within I2\mathcal{I}^2 into nn subintervals:

Wn(u,v)=max(k1n,u+vkn)\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)[k1n,kn]×[k1n,kn]\mbox, k=1,2,,n(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

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

The support of Wn\mathbf{W}_n consists of nn line segments connecting coordinate pairs {(k1)/n,k/n}\{(k-1)/n,\, k/n\} and {k/n,(k1)/n}\{k/n,\, (k-1)/n\} as stated by Nelsen (2006). The Spearman Rho (rhoCOP) is defined by ρC=1(2/n2)\rho_\mathbf{C} = 1 - (2/n^2), and the Kendall Tau (tauCOP) by τC=1(2/n)\tau_\mathbf{C} = 1 - (2/n).

Usage

W_N5p12a(u, v, para=1, ...)

Arguments

u

Nonexceedance probability uu in the XX direction;

v

Nonexceedance probability vv in the YY direction;

para

A positive integer n1,2,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)

[Package copBasic version 2.2.4 Index]