lmomco (version 2.4.14)

Lcomoment.Wk: Weighting Coefficient for Sample L-comoment

Description

Compute the weight factors for computation of an L-comoment for order k, order statistic r, and sample size n.

Usage

Lcomoment.Wk(k,r,n)

Value

A single L-comoment weight factor.

Arguments

k

Order of L-comoment being computed by parent calls to Lcomoment.Wk.

r

Order statistic index involved.

n

Sample size.

Author

W.H. Asquith

Details

This function computes the weight factors needed to calculation L-comoments and is interfaced or used by Lcomoment.Lk12. The weight factors are $$ w^{(k)}_{r:n} = \sum_{j=0}^{min\{r-1,k-1\}} (-1)^{k-1-j} \frac{{k-1 \choose j}{k-1+j \choose j} {r-1 \choose j}} {{n-1 \choose j}} \mbox{.}$$ The weight factor \(w^{(k)}_{r:n}\) is the discrete Legendre polynomial. The weight factors are well illustrated in figure 6.1 of Asquith (2011). This function is not intended for end users.

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.Lk12

Examples

Wk <- Lcomoment.Wk(2,3,5)
print(Wk)

if (FALSE) {
# To compute the weight factors for L-skew and L-coskew (k=3) computation
# for a sample of size 20.
Wk <- matrix(nrow=20,ncol=1)
for(r in seq(1,20)) Wk[r] <- Lcomoment.Wk(3,r,20)
plot(seq(1,20),Wk, type="b")
}

# The following shows the actual weights used for computation of
# the first four L-moments. The sum of the each sample times the
# corresponding weight equals the L-moment.
fakedat <- sort(c(-10, 20, 30, 40));  n <- length(fakedat)
Wk1 <- Wk2 <- Wk3 <- Wk4 <- vector(mode="numeric", length=n);
for(i in 1:n) {
   Wk1[i] <- Lcomoment.Wk(1,i,n)/n
   Wk2[i] <- Lcomoment.Wk(2,i,n)/n
   Wk3[i] <- Lcomoment.Wk(3,i,n)/n
   Wk4[i] <- Lcomoment.Wk(4,i,n)/n
}
cat(c("Weights for mean",         round(Wk1, digits=4), "\n"))
cat(c("Weights for L-scale",      round(Wk2, digits=4), "\n"))
cat(c("Weights for 3rd L-moment", round(Wk3, digits=4), "\n"))
cat(c("Weights for 4th L-moment", round(Wk4, digits=4), "\n"))
my.lams <- c(sum(fakedat*Wk1), sum(fakedat*Wk2),
             sum(fakedat*Wk3), sum(fakedat*Wk4))
cat(c("Manual L-moments:", my.lams, "\n"))
cat(c("lmomco L-moments:", lmoms(fakedat, nmom=4)$lambdas,"\n"))
# The last two lines of output should be the same---note that lmoms()
# does not utilize Lcomoment.Wk(). So a double check is made.