Hosking and Wallis sample L-moments

Description

Lmoments provides the estimate of L-moments of a sample or regional L-moments of a region.

Usage

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 Lmoments (x)
 regionalLmoments (x,cod)
 LCV (x)
 LCA (x)
 Lkur (x)

Arguments

x

vector representing a data-sample (or data from many samples defined with cod in the case of regionalLmoments)

cod

array that defines the data subdivision among sites

Details

The estimation of L-moments is based on a sample of size n, arranged in ascending order. Let x(1:n) <= x(2:n) <= ... <= x(n:n) be the ordered sample. An unbiased estimator of the probability weighted moments βr is:

br = 1/n sum[j from r+1 to n](x(j:n) (j-1)(j-2)...(j-r)/(n-1)/(n-2)/.../(n-r))

The sample L-moments are defined by:

l1 = b0

l2 = 2b1 - b0

l3 = 6b2 - 6b1 + b0

l4 = 20b3 - 30b2 + 12b1 - b0

and in general

l(r+1) = sum[k from 0 to r](b_k (-1)^(r-k) (r+k)! / (k!)^2 / (r-k)!)

where r=0, 1, ..., n-1.

The sample L-moment ratios are defined by

tr = lr / l2

and the sample L-CV by

t = l2 / l1

Sample regional L-CV, L-skewness and L-kurtosis coefficients are defined as

t^R = sum[i from 1 to k](ni t^(i)) / sum[i from 1 to k](ni)

t3^R = sum[i from 1 to k](ni t3^(i)) / sum[i from 1 to k](ni)

t4^R = sum[i from 1 to k](ni t4^(i)) / sum[i from 1 to k](ni)

Value

Lmoments gives the L-moments (l1, l2, t, t3, t4), regionalLmoments gives the regional weighted L-moments (l1^R, l2^R, t^R, t3^R, t4^R), LCV gives the coefficient of L-variation, LCA gives the L-skewness and Lkur gives the L-kurtosis of x.

Author(s)

Alberto Viglione, e-mail: alviglio@tiscali.it.

References

Hosking, J.R.M. and Wallis, J.R. (1997) Regional Frequency Analysis: an approach based on L-moments, Cambridge University Press, Cambridge, UK.

See Also

mean, var, sd, HOMTESTS.

Examples

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x <- rnorm(30,10,2)
Lmoments(x)

data(annualflows)
annualflows
summary(annualflows)
x <- annualflows["dato"][,]
cod <- annualflows["cod"][,]
split(x,cod)
camp <- split(x,cod)$"45"
Lmoments(camp)
sapply(split(x,cod),Lmoments)

regionalLmoments(x,cod)