ORDSUMcop: Ordinal Sums of M-Copula

Description

Compute ordinal sums of a copula (Nelsen, 2006, p. 63) or M-ordinal sum of the summands (Klement et al., 2017) within $\mathcal{I}^2$ into $n$ partitions (possibly infinite) within $\mathcal{I}^2$. According to Nelsen, letting $\mathcal{J}$ denote a partition of $\mathcal{I}^2$ and $\mathcal{J}_i = [a_i,\, b_i]$ be the $i$th partition that does not overlap with others and letting also $\mathbf{C}_i$ be a copula for the $i$th partition, then the ordinal sum of these $\mathbf{C}_i$ with parameters $\Theta_i$ with respect to $\mathcal{J}_i$ is the copula $\mathbf{C}$ given by

$$\mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, \Theta_i, i \in 1,2,\cdots,n\bigr) = a_i + (b_i-a_i)\mathbf{C}_i\biggl(\frac{u-a_i}{b_i-a_i},\, \frac{v-a_i}{b_i-a_i}; \Theta_i\biggr)\ \mbox{for}\ (u,v) \in \mathcal{J}^2\mbox{,}$$

for points within the partitions, and for points otherwise outside the partitions the coupla is given by

$$\mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, i \in 1,2,\cdots,n\bigr) = \mathbf{M}(u,v)\ \mathrm{for}\ (u,v) \ni \mathcal{J}^2\mbox{, and}$$

let $\mathbf{C}_\mathcal{J}(u,v)$ be a convenient abbreviation for the copula. Finally, Nelsen (2006, theorem 3.2.1) states that a copula is an ordinal sum if and only if for a $t$ if $\mathbf{C}(t,t)=t$ for $t \in (0,1)$. The diagonal of a coupla can be useful for quick assessment (see Examples) of this theorem. (See ORDSUWcop, W-ordinal sum of the summands.)

Usage

ORDSUMcop(u,v, para=list(cop=W, para=NA, part=c(0,1)), ...)

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 list of sublists for the coupla, parameters, and partitions (see Examples) and some attempt for intelligent in-fill of para is made within the sources (the default para is an example for which cop and para elements are converted to lists). The user is responsible that part element properly canvases by end-point alignment all of $\mathcal{I}^2$; and
...: Additional arguments to pass.

Author

W.H. Asquith

References

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