Lmoments {nsRFA}

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

 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} \le x_{2:n} \le \dots \le x_{n:n} be the ordered sample. An unbiased estimator of the probability weighted moments \beta_r is:

b_r = n^{-1} \sum_{j=r+1}^n \frac{(j-1)(j-2)\dots(j-r)}{(n-1)(n-2)\dots(n-r)} x_{j:n}

The sample L-moments are defined by:

l_1 = b_0

l_2 = 2b_1 - b_0

l_3 = 6b_2 - 6b_1 + b_0

l_4 = 20b_3-30b_2+12b_1-b_0

and in general

l_{r+1} = \sum_{k=0}^r \frac{(-1)^{r-k}(r+k)!}{(k!)^2(r-k)!} b_k

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

The sample L-moment ratios are defined by

t_r=l_r/l_2

and the sample L-CV by

t=l_2/l_1

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

t^R = \frac{\sum_{i=1}^k n_i t^{(i)}}{ \sum_{i=1}^k n_i}

t_3^R =\frac{ \sum_{i=1}^k n_i t_3^{(i)}}{ \sum_{i=1}^k n_i}

t_4^R =\frac{ \sum_{i=1}^k n_i t_4^{(i)}}{\sum_{i=1}^k n_i}

Value

Lmoments gives the L-moments (l_1, l_2, t, t_3, t_4), regionalLmoments gives the regional weighted L-moments (l_1^R, l_2^R, t^R, t_3^R, t_4^R), LCV gives the coefficient of L-variation, LCA gives the L-skewness and Lkur gives the L-kurtosis of x.

Note

For information on the package and the Author, and for all the references, see nsRFA.

See Also

mean, var, sd, HOMTESTS.

Examples

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

data(hydroSIMN)
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)