kfuncCOPlmoms: The L-moments of the Kendall Function of a Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

The L-moments of the Kendall Function of a Copula

Description

Compute the L-moments of the Kendall Function (F_K(z; \mathbf{C})) of a copula \mathbf{C}(u,v) where the z is the joint probability of the \mathbf{C}(u,v). The Kendall Function (or Kendall Distribution Function) is the cumulative distribution function (CDF) of the joint probability Z of the coupla. The expected value of the z(F_K) (mean, first L-moment \lambda_1), because Z has nonzero probability for 0 \le Z \le \infty, is

\mathrm{E}[Z] = \lambda_1 = \int_0^\infty \bigl[1 - F_K(t)\bigr]\,\mathrm{d}t = \int_0^1 \bigl[1 - F_K(t)\bigr] \,\mathrm{d}t\mbox{,}

where for circumstances here 0 \le Z \le 1. The \infty is mentioned only because expectations of such CDFs are usually shown using (0,\infty) limits, whereas integration of quantile functions (CDF inverses) use limits (0, 1). Because the support of Z is (0, 1), like the probability F_K, showing just it (\infty) as the upper limit could be confusing—statements such as “probabilities of probabilities” are rhetorically complex. So, pursuit of word precision is made herein.

An expression for \lambda_r for r \ge 2 in terms of the F_K(z) is

\lambda_r = \frac{1}{r}\sum_{j=0}^{r-2} (-1)^j {r-2 \choose j}{r \choose j+1} \int_{0}^{1} \! \bigl[F_K(t)\bigr]^{r-j-1}\times \bigl[1 - F_K(t)\bigr]^{j+1}\, \mathrm{d}t\mbox{,}

where because of these circumstances the limits of integration are (0, 1) and not (-\infty, \infty) as in the usual definition of L-moments in terms of a distribution's CDF. (Note, such expressions did not make it into Asquith (2011), which needs rectification if that monograph ever makes it to a 2nd edition.)

The mean, L-scale, coefficient of L-variation (\tau_2, LCV, L-scale/mean), L-skew (\tau_3, TAU3), L-kurtosis (\tau_4, TAU4), and \tau_5 (TAU5) are computed. In usual nomenclature, the L-moments are \lambda_1 = \mbox{mean,} \lambda_2 = \mbox{L-scale,} \lambda_3 = \mbox{third L-moment,} \lambda_4 = \mbox{fourth L-moment, and} \lambda_5 = \mbox{fifth L-moment,} whereas the L-moment ratios are \tau_2 = \lambda_2/\lambda_1 = \mbox{coefficient of L-variation, } \tau_3 = \lambda_3/\lambda_2 = \mbox{L-skew, } \tau_4 = \lambda_4/\lambda_2 = \mbox{L-kurtosis, and} \tau_5 = \lambda_5/\lambda_2 = \mbox{not named.} It is common amongst practitioners to lump the L-moment ratios into the general term “L-moments” and remain inclusive of the L-moment ratios. For example, L-skew then is referred to as the 3rd L-moment when it technically is the 3rd L-moment ratio. There is no first L-moment ratio (meaningless); so, results from kfuncCOPlmoms function will canoncially show a NA in that slot. The coefficient of L-variation is \tau_2 (subscript 2) and not Kendall Tau (\tau). Sample L-moments are readily computed by several packages in R (e.g. lmomco, lmom, Lmoments, POT).

Usage

kfuncCOPlmom(r, cop=NULL, para=NULL, ...)

kfuncCOPlmoms(cop=NULL, para=NULL, nmom=5, begin.mom=1, ...)

Arguments

r	The rth order of a single L-moment to compute;
cop	A copula function;
para	Vector of parameters or other data structure, if needed, to pass to the copula;
nmom	The number of L-moments to compute;
begin.mom	The rth order to begin the sequence lambegr:nmom for L-moment computation. The rarely used argument is means to bypass the computation of the mean if the user has an alternative method for the mean or other central tendency characterization in which case begin.mom = 2; and
...	Additional arguments to pass.

Value

An R list is returned by kfuncCOPlmoms and only the scalar value of \lambda_r by kfuncCOPlmom.

lambdas	Vector of the L-moments. First element is \lambda_1, second element is \lambda_2, and so on;
ratios	Vector of the L-moment ratios. Second element is \tau, third element is \tau_3 and so on; and
source	An attribute identifying the computational source of the L-moments: “kfuncCOPlmoms”.

Note

The L-moments of Kendall Functions appear to be not yet fully researched. An interesting research direction would be the trajectories of the L-moments or L-moment ratio diagrams for the Kendall Function and the degree to which distinction between copulas becomes evident—such diagrams are in wide-spread use for distinquishing between univariate distributions. It is noted, however, that Kendall Function L-moment ratio diagrams might be of less utility that in the univariate world—recalling that a univariate distribution is unique characteristized by its L-moments—because different copulas can have the same F_K(z), such as all bivariate extreme value copulas (see also Examples).

  Rhos <- c(0.001, 0.01, seq(0.05, 0.95, by=0.05), 0.99, 0.999)
  L1 <- T2 <- T3 <- T4 <- Thetas <- vector(mode="numeric", length(Rhos))
  for(i in 1:length(Thetas)) {
     Thetas[i] <- uniroot(function(p)
                 Rhos[i] - rhoCOP(cop=PARETOcop, para=p), c(0,200))$root
     message("Rho = ", Rhos[i], " and Pareto theta = ",
                                             round(Thetas[i], digits=4))
     lmr <- kfuncCOPlmoms(cop=PARETOcop, para=Thetas[i], nmom=4)
     L1[i] <- lmr$lambdas[1]; T2[i] <- lmr$ratios[2]
     T3[i] <- lmr$ratios[3];  T4[i] <- lmr$ratios[4]
  }
  LMR <- data.frame(Rho=Rhos, Theta=Thetas, L1=L1, T2=T2, T3=T3, T4=T4)
  plot(LMR$Rho, LMR$T2, ylim=c(-0.04, 0.5), xlim=c(0, 1),
       xlab="Spearman Rho or coefficient of L-variation",
       ylab="L-moment ratio", type="l", col="black")
  lines(LMR$Rho, LMR$T3, lty=1, col="red"         )
  lines(LMR$Rho, LMR$T4, lty=1, col="green"       )
  lines(LMR$T2,  LMR$T3, lty=2, col="blue"        )
  lines(LMR$T2,  LMR$T4, lty=2, col="deepskyblue2")
  lines(LMR$T3,  LMR$T4, lty=2, col="purple"      )

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

See Also

kfuncCOP

Examples

## Not run: 
kfuncCOPlmom(1, cop=P) # 0.5 * 0.5 = 0.25 is expected joint prob. of independence
#[1] 0.2499999  (in agreement with theory)

ThetaGH <- 4.21
Rho <- rhoCOP(cop=GHcop, para=ThetaGH)
ThetaHR <- uniroot(function(p) Rho - rhoCOP(cop=HRcop, para=p), c(0, 100))$root
ThetaHR <- uniroot(function(p) Rho - rhoCOP(cop=HRcop, para=p), c(0, 100))$root
ThetaGL <- uniroot(function(p) Rho - rhoCOP(cop=GLcop, para=p), c(0, 100))$root
ls.str(kfuncCOPlmoms(cop=GHcop, para=ThetaGH)) # Gumbel-Hougaard copula
# lambdas :  num [1:5] 0.440617 0.169085 0.011228 -0.000797 0.000249
# ratios  :  num [1:5]       NA 0.383750 0.066400 -0.004720 0.001470
#                               L-skew = 0.066400
ls.str(kfuncCOPlmoms(cop=HRcop, para=ThetaHR)) # Husler-Reiss copula
# lambdas :  num [1:5] 0.439627 0.169052 0.011427 -0.000785 0.000249
# ratios  :  num [1:5]       NA 0.384540 0.067590 -0.004640 0.001470
#                               L-skew = 0.067590
ls.str(kfuncCOPlmoms(cop=GLcop, para=ThetaGL)) # Galambos copula
# lambdas :  num [1:5] 0.440415 0.169079 0.011268 -0.000795 0.000248
# ratios  :  num [1:5]       NA 0.383910 0.066650 -0.004700 0.001470
#                               L-skew = 0.066650
# These L-moments are extremely similar and within the numerics used.
# Extreme value copula all have the same Kendall Distribution function.
## End(Not run)

## Not run: 
UV <- simCOP(200, cop=PLcop, para=1/pi, graphics=FALSE)
theta <- PLpar(UV[,1], UV[,2])
zs <- c(0.001, seq(0.01, 0.99, by=0.01), 0.999) # for later

# Take the sample estimated parameter and convert to joint probabilities Z
# Convert the Z to the Kendall Function estimates again with the sample parameter
Z  <- PLcop(UV[,1], UV[,2], para=theta); KF <- kfuncCOP(Z, cop=PLcop, para=theta)

# Compute L-moments of the "Kendall function" and the sample versions
# and again see that the L-moment are for the distribution of the Z!
KNFlmr <- kfuncCOPlmoms(cop=PLcop, para=theta); SAMlmr <- lmomco::lmoms(Z)
knftxt <- paste0("Kendall L-moments: ",
                 paste(round(KNFlmr$lambdas, digits=4), collapse=", "))
samtxt <- paste0("Sample L-moments: " ,
                 paste(round(SAMlmr$lambdas, digits=4), collapse=", "))

plot(Z, KF, xlim=c(0,1), ylim=c(0,1), col="black",
     xlab="COPULA(u,v) VALUE [JOINT PROBABILITY]",
     ylab="KENDALL DISTRIBUTION FUNCTION (KDF), AS NONEXCEEDANCE PROBABILITY")
rug(Z, side=1, col="red", lwd=0.5); rug(KF, side=2, col="red", lwd=0.5) # rug plots
lines(zs, kfuncCOP(zs, cop=PLcop, para=1/pi), col="darkgreen")
knf_meanZ <- KNFlmr$lambdas[1]; sam_meanZ <- SAMlmr$lambdas[1]
knf_mean  <- kfuncCOP(knf_meanZ, cop=PLcop, para=theta) # theo. Kendall function
sam_mean  <- kfuncCOP(sam_meanZ, cop=PLcop, para=theta) # sam. est. of Kendall func
points(knf_meanZ, knf_mean, pch=16, col="blue", cex=3)
points(sam_meanZ, sam_mean, pch=16, col="cyan", cex=2)
lines(zs, zs-zs*log(zs), lty=2, lwd=0.8) # dash ref line for independence
text(0.2, 0.30, knftxt, pos=4, cex=1); text(0.2, 0.25, samtxt, pos=4, cex=1)
text(0.2, 0.18, paste0("Notice uniform distribution of vertical axis rug.\n",
                       "A Critical remark with respect to to KDFs."), cex=1, pos=4)
legend("bottomright", c("Independence copula", "KDF of Plackett copula",
                        "Theoretical mean", "Sample mean"), bty="n", y.intersp=1.5,
       lwd=c(1, 1, NA, NA), lty=c(2, 1, NA, NA), pch=c(NA, NA, 16, 16),
       col=c("black", "darkgreen", "blue", "cyan"), pt.cex=c(NA, NA, 3, 2)) #
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.