copBasic (version 2.1.4)

PLACKETTpar: Estimate the Parameter of the Plackett Copula

Description

The parameter \(\Theta\) of the Plackett copula (Nelsen, 2006, pp. 89--92) 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's Rho (see rhoCOP) is provided by rho.

Usage

PLACKETTpar(x, y, rho=NULL, byrho=FALSE, cor=NULL, ...)
      PLpar(x, y, rho=NULL, byrho=FALSE, cor=NULL, ...)

Arguments

x

Vector of values for random variable \(X\);

y

Vector of values for random variable \(Y\);

rho

Spearman Rho and byrho is set to TRUE automatically;

byrho

Should Spearman Rho be used instead of the median-split estimator;

cor

A copBasic syntax for “the correlation coefficient” suitable for the copula---a synonym for rho; and

...

Additional arguments to pass.

Value

A value for the Plackett copula \(\Theta\) is returned.

References

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.

See Also

PLACKETTcop, PLACKETTsim, rhoCOP

Examples

# NOT RUN {
Q1 <- rnorm(1000); Q2 <- Q1+rnorm(1000)
PLpar(Q1,Q2); PLpar(Q1,Q2, byrho=TRUE) # two estimates for same data
PLpar(rho= 0.76) # positive association
PLpar(rho=-0.76) # negative association
tauCOP(cop=PLcop, para=PLpar(rho=-0.15, by.rho=TRUE))

# }
# NOT RUN {
RHOS <- seq(0,0.990, by=0.002); TAUS <- rep(NA, length(RHOS))
for(i in 1:length(RHOS)) {
   #message("Spearman Rho: ", RHOS[i])
   theta <- PLACKETTpar(rho=RHOS[i], by.rho=TRUE); tau <- NA
   try(tau <- tauCOP(cop=PLACKETTcop, para=theta), silent=TRUE)
   TAUS[i] <- ifelse(is.null(tau), NA, tau)
}
LM <- lm(TAUS~  RHOS    + I(RHOS^2) + I(RHOS^3) + I(RHOS^4) +
              I(RHOS^5) + I(RHOS^6) + I(RHOS^7) + I(RHOS^8) - 1)
plot(RHOS,TAUS, type="l", xlab="abs(Spearman Rho)", ylab="abs(Kendall Tau)")
lines(RHOS,fitted.values(LM), col=3)#
# }