Lmoments provides the estimate of L-moments of a sample or regional L-moments of a region.
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vector representing a data-sample (or data from many samples defined with
array that defines the data subdivision among sites
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)
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
Alberto Viglione, e-mail: [email protected].
Hosking, J.R.M. and Wallis, J.R. (1997) Regional Frequency Analysis: an approach based on L-moments, Cambridge University Press, Cambridge, UK.
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