Lcomoment.Wk {lmomco}

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)

Arguments

k

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

r

Order statistic index involved.

n

Sample size.

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.

Value

A single L-comoment weight factor.

Note

The function begins with a capital letter. This is intentionally done so that lower case namespace is preserved. By using a capital letter now, then lcomoment.Wk remains an available name in future releases.

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.

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)

## Not run: 
# 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")

## End(Not run)

# 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.