lmomco (version 2.4.14)

Lcomoment.correlation: L-correlation Matrix (L-correlation through Sample L-comoments)

Description

Compute the L-correlation from an L-comoment matrix of order \(k = 2\). This function assumes that the 2nd order matrix is already computed by the function Lcomoment.matrix.

Usage

Lcomoment.correlation(L2)

Value

An R

list is returned.

type

The type of L-comoment representation in the matrix: “Lcomoment.coefficients”.

order

The order of the matrix---extracted from the first matrix in arguments.

matrix

A \(k \ge 2\) L-comoment coefficient matrix.

Arguments

L2

A \(k = 2\) L-comoment matrix from Lcomoment.matrix(Dataframe,k=2).

Author

W.H. Asquith

Details

L-correlation is computed by Lcomoment.coefficients(L2,L2) where L2 is second order L-comoment matrix. The usual L-scale values as seen from lmom.ub or lmoms are along the diagonal. This function does not make use of lmom.ub or lmoms and can be used to verify computation of \(\tau\) (coefficient of L-variation).

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.

Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments---L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765--1781.

See Also

Lcomoment.matrix, Lcomoment.correlation

Examples

D   <- data.frame(X1=rnorm(30), X2=rnorm(30), X3=rnorm(30))
L2  <- Lcomoment.matrix(D,k=2)
RHO <- Lcomoment.correlation(L2)
if (FALSE) {
"SerfXiao.eq17" <-
 function(n=25, A=10, B=2, k=4,
          method=c("pearson","lcorr"), wrt=c("12", "21")) {
   method <- match.arg(method); wrt <- match.arg(wrt)
   # X1 is a linear regression on X2
   X2 <- rnorm(n); X1 <- A + B*X2 + rnorm(n)
   r12p <- cor(X1,X2) # Pearson's product moment correlation
   XX <- data.frame(X1=X1, X2=X2) # for the L-comoments
   T2 <- Lcomoment.correlation(Lcomoment.matrix(XX, k=2))$matrix
   LAMk <- Lcomoment.matrix(XX, k=k)$matrix # L-comoments of order k
   if(wrt == "12") { # is X2 the sorted variable?
      lmr <- lmoms(X1, nmom=k); Lamk <- LAMk[1,2]; Lcor <- T2[1,2]
   } else {          # no X1 is the sorted variable (21)
      lmr <- lmoms(X2, nmom=k); Lamk <- LAMk[2,1]; Lcor <- T2[2,1]
   }
   # Serfling and Xiao (2007, eq. 17) state that
   # L-comoment_k[12] = corr.coeff * Lmoment_k[1] or
   # L-comoment_k[21] = corr.coeff * Lmoment_k[2]
   # And with the X1, X2 setup above, Pearson corr. == L-corr.
   # There will be some numerical differences for any given sample.
   ifelse(method == "pearson",
             return(lmr$lambdas[k]*r12p - Lamk),
             return(lmr$lambdas[k]*Lcor - Lamk))
   # If the above returns a expected value near zero then, their eq.
   # is numerically shown to be correct and the estimators are unbiased.
}

# The means should be near zero.
nrep <- 2000; seed <- rnorm(1); set.seed(seed)
mean(replicate(n=nrep, SerfXiao.eq17(method="pearson", k=4)))
set.seed(seed)
mean(replicate(n=nrep, SerfXiao.eq17(method="lcorr", k=4)))
# The variances should nearly be equal.
seed <- rnorm(1); set.seed(seed)
var(replicate(n=nrep, SerfXiao.eq17(method="pearson", k=6)))
set.seed(seed)
var(replicate(n=nrep, SerfXiao.eq17(method="lcorr", k=6)))
}