Lmoments {nsRFA} | R Documentation

## 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) <= 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 (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)

[Package nsRFA version 0.6-9 Index]