The parameter \(\Theta\) of the Plackett copula (Nelsen, 2006, pp. 89--92) (PLACKETTcop or PLcop) is related to the Spearman Rho (\(\rho_S \ne 1\), see rhoCOP)
$$\rho_S(\Theta) = \frac{\Theta + 1}{\Theta - 1} - \frac{2\Theta\log(\Theta)}{(\Theta - 1)^2}\mbox{.}$$
Alternatively, the parameter can be estimated using a median-split estimator, which is best shown as an algorithm. First, compute the two medians:
medx <- median(x); medy <- median(y)
Second and third, compute the number of occurrences where both values are less than their medians and express that as a probability:
k <- length(x[x < medx & y < medy]); m <- k / length(x)
Finally, the median-split estimator of \(\Theta\) is computed by
$$\Theta = \frac{4m^2}{(1-2m)^2}\mbox{.}$$
Nelsen (2006, p. 92) and Salvadori et al. (2007, p. 247) provide further details. The input values x and y are not used for the median-split estimator if Spearman Rho (see rhoCOP) is provided by rho.
PLACKETTpar(x, y, rho=NULL, byrho=FALSE, cor=NULL, ...)
PLpar(x, y, rho=NULL, byrho=FALSE, cor=NULL, ...)A value for the Plackett copula \(\Theta\) is returned.
Vector of values for random variable \(X\);
Vector of values for random variable \(Y\);
Spearman Rho and byrho is set to TRUE automatically;
Should Spearman Rho be used instead of the median-split estimator;
A copBasic syntax for “the correlation coefficient” suitable for the copula---a synonym for rho; and
Additional arguments to pass.
W.H. Asquith
Fredricks, G.A, and Nelsen, R.B., 2007, On the relationship between Spearman's rho and Kendall's tau for pairs of continuous random variables: Journal of Statistical Planning and Inference, v. 137, pp. 2143--2150.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.
PLACKETTcop, PLcop, PLACKETTsim, rhoCOP