Compute the primary diagonal or alternatively the secondary diagonal (Nelsen, 2006, pp. 12 and 16) of copula \(\mathbf{C}(u,v)\).
The primary diagonal is defined as
$$\mathbf{\delta}_\mathbf{C}(t) = \mathbf{C}(t,t)\mbox{,}$$
and the secondary diagonal is defined as
$$\mathbf{\delta}^{\star}_\mathbf{C}(t) = \mathbf{C}(t,1-t)\mbox{.}$$
Plotting is provided by this function because the diagonals are such important visual attributes of a copula. This function computes whole diagonals. If individual values are desired, then users are asked to use function calls along the diagonal such as COP(0.25,0.25, cop=P) for the primary diagonal and COP(0.25,1-0.25, cop=P) for the secondary diagonal, where for both examples the independence copula (\(uv = \mathbf{\Pi}\); P) was chosen for purposes of clarification.
The \(\mathbf{\delta}_\mathbf{C}(t)\) is related to order statistics of the multivariate sample (here bivariate) (Durante and Sempi, 2015, p. 68). The probability for the maxima is \(\mathrm{Pr}[\mathrm{max}(u, v) \le t] = \mathbf{C}(t,t) = \mathbf{\delta}_\mathbf{C}(t) \mbox{\ and }\) the probability for the minima is \(\mathrm{Pr}[\mathrm{min}(u, v) \le t] = 2t - \mathbf{\delta}_\mathbf{C}(t)\mbox{.}\)
diagCOP(cop=NULL, para=NULL, secondary=FALSE,
ploton=TRUE, lines=TRUE, delt=0.005, ...)An R
list of the \(t\) values, \(\mathbf{\delta}_\mathbf{C}(t,t)\) (primary) or \(\mathbf{\delta}^{\star}_\mathbf{C}(t,1-t)\) (secondary diagonal), along with a tag as to which diagonal is returned.
A copula function;
Vector of parameters, if needed, to pass to the copula;
A logical to toggle the secondary diagonal;
A logical to toggle on the plot;
Draw the lines of diagonal to the current device;
The increment of the diagonal curve to plot, defaults to 0.5-percent intervals, which should be small enough to resolve fine curvature for many copulas in practice; and
Additional arguments to pass to the plot() and lines() functions in R.
W.H. Asquith
Durante, F., and Sempi, C., 2015, Principles of copula theory: Boca Raton, CRC Press, 315 p.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
diagCOPatf, COP, sectionCOP