# lmom2pwm: L-moments to Probability-Weighted Moments In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

Converts the L-moments to the probability-weighted moments (PWMs) given the L-moments. The conversion is linear so procedures based on L-moments are identical to those based on PWMs. The expression linking PWMs to L-moments is

λ_{r+1} = ∑_{k=0}^r (-1)^{r-k} {r \choose k}{r+k \choose k}β_k\mbox{,}

where λ_{r+1} are the L-moments, β_r are the PWMs, and r ≥ 0.

## Usage

 `1` ```lmom2pwm(lmom) ```

## Arguments

 `lmom` An L-moment object created by `lmoms`, `lmom.ub`, or `vec2lmom`. The function also supports `lmom` as a vector of L-moments (λ_1, λ_2, τ_3, τ_4, and τ_5).

## Details

PWMs are linear combinations of the L-moments and therefore contain the same statistical information of the data as the L-moments. However, the PWMs are harder to interpret as measures of probability distributions. The PWMs are included in lmomco for theoretical completeness and are not intended for use with the majority of the other functions implementing the various probability distributions. The relations between L-moments (λ_r) and PWMs (β_{r-1}) for 1 ≤ r ≤ 5 order are

λ_1 = β_0 \mbox{,}

λ_2 = 2β_1 - β_0 \mbox{,}

λ_3 = 6β_2 - 6β_1 + β_0 \mbox{,}

λ_4 = 20β_3 - 30β_2 + 12β_1 - β_0\mbox{, and}

λ_5 = 70β_4 - 140β_3 + 90β_2 - 20β_1 + β_0\mbox{.}

The linearity between L-moments and PWMs means that procedures based on one are equivalent to the other. This function only accomodates the first five L-moments and PWMs. Therefore, at least five L-moments are required in the passed argument.

## Value

An R `list` is returned.

 `betas` The PWMs. Note that convention is the have a β_0, but this is placed in the first index `i=1` of the `betas` vector. `source` Source of the PWMs: “pwm”.

W.H. Asquith

## References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979, Probability weighted moments—Definition and relation to parameters of several distributions expressable in inverse form: Water Resources Research, v. 15, pp. 1,049–1,054.

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124.

`lmom.ub`, `lmoms`, `pwm.ub`, `pwm2lmom`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```pwm <- lmom2pwm(lmoms(c(123,34,4,654,37,78))) lmom2pwm(lmom.ub(rnorm(100))) lmom2pwm(lmoms(rnorm(100))) lmomvec1 <- c(1000,1300,0.4,0.3,0.2,0.1) pwmvec <- lmom2pwm(lmomvec1) print(pwmvec) #\$betas #[1] 1000.0000 1150.0000 1070.0000 984.5000 911.2857 # #\$source #[1] "lmom2pwm" lmomvec2 <- pwm2lmom(pwmvec) print(lmomvec2) #\$lambdas #[1] 1000 1300 520 390 260 # #\$ratios #[1] NA 1.3 0.4 0.3 0.2 # #\$source #[1] "pwm2lmom" pwm2lmom(lmom2pwm(list(L1=25, L2=20, TAU3=.45, TAU4=0.2, TAU5=0.1))) ```