PLACKETTpar {copBasic}

Estimate the Parameter of the Plackett Copula

Description

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.

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.

Note

Evidently there “does not appear to be a closed form for \tau(\Theta)” (Fredricks and Nelsen, 2007, p. 2147), but given \rho(\Theta), the equivalent \tau(\Theta) can be computed by either the tauCOP function or by approximation. One of the Examples sweeps through \rho \mapsto [0,1; \Delta\rho{=}\delta], fits the Plackett \theta(\rho), and then solves for Kendall Tau \tau(\theta) using tauCOP. A polynomial is then fit between \tau and \rho to provide rapid conversion between |\rho| and \tau, where the residual standard error is 0.0005705, adjusted R-squared is \approx 1, the maximum residual is \epsilon < 0.006. Because of symmetry, it is only necessary to fit positive association and reflect the result by the sign of \rho. This polynomial is from the Examples is

  rho <- 0.920698
  "getPLACKETTtau" <- function(rho) {
     taupoly <-   0.6229945*abs(rho)   +   1.1621854*abs(rho)^2 -
                 10.7424188*abs(rho)^3 +  48.9687845*abs(rho)^4 -
                119.0640264*abs(rho)^5 + 160.0438496*abs(rho)^6 -
                111.8403591*abs(rho)^7 +  31.8054602*abs(rho)^8
     return(sign(rho)*taupoly)
  }
  getPLACKETTtau(rho) # 0.7777726

The following code might be useful in some applications for the inversion of the polynomial for the \rho as a function of \tau:

  "fun" <- function(rho, tau=NULL) {tp <- getPLACKETTtau(rho); return(tau-tp)}
   tau  <- 0.78
   rho  <- uniroot(fun, interval=c(0, 1), tau=tau)$root # 0.9220636

Author(s)

W.H. Asquith

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, PLcop, 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)) #
## End(Not run)

## 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)#
## End(Not run)