# 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

1 2 3 4 5 | ```
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` |
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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
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)
``` |