nsRFA (version 0.7-15)

Lmoments: 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

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.

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 $$t^R = \frac{\sum_{i=1}^k n_i t^{(i)}}{ \sum_{i=1}^k n_i}$$ $$t_3^R =\frac{ \sum_{i=1}^k n_i t_3^{(i)}}{ \sum_{i=1}^k n_i}$$ $$t_4^R =\frac{ \sum_{i=1}^k n_i t_4^{(i)}}{\sum_{i=1}^k n_i}$$

See Also

mean, var, sd, HOMTESTS.

Examples

# NOT RUN {
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)
# }