PLACKETTpar: Estimate the Parameter of the Plackett Copula
Description
The parameter $\Theta$ of the Plackett copula (Nelson, 2006, pp. 89--92) is related to Spearman's Rho ($\rho_S \ne 1$, see rhoCOP)
$$\rho_S(\Theta) = \frac{\Theta + 1}{\Theta - 1} - \frac{2\Theta\log(\Theta)}{(\Theta - 1)^2}$$
and a median-split estimator best shown as an algorithm. First, compute the two medians:
medx <- median(x) and medy <- median(y). Second, compute the number of occurrences where both values are less than their medians k <- length(x[x < medx & y < medy]). Third, express this as a probability m <- k/length(x), and finally the median-split estimator of $\Theta$ computed by
$$\Theta = \frac{4m^2}{(1-2m)^2}\mbox{.}$$
Nelson (2006, p. 92) and Salvadori et al. (2007, p. 247) provide further details. The input values x and y are not used if Spearman's Rho is provided by rho.
Usage
PLACKETTpar(x, y, rho=NULL, byrho=FALSE, ...)
Arguments
x
Vector of values for random variable $X$;
y
Vector of values for random variable $Y$;
rho
Spearman's Rho and byrho is set to TRUE automatically;
byrho
Should Spearman's Rho be used instead of the median-split estimator; and
...
Additional arguments to pass.
Value
A value for the Plackett copula $\Theta$ is returned.
References
Nelson, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.