# pwm2lmom: Probability-Weighted Moments to L-moments In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

Converts the probability-weighted moments (PWM) to the L-moments. The conversion is linear so procedures based on PWMs are identical to those based on L-moments through a system of linear equations

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{,}

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

τ = λ_2/λ_1 \mbox{,}

τ_3 = λ_3/λ_2 \mbox{,}

τ_4 = λ_4/λ_2 \mbox{, and}

τ_5 = λ_5/λ_2 \mbox{.}

The general expression and the expression used for computation if the argument is a vector of PWMs is

λ_{r+1} = ∑^r_{k=0} (-1)^{r-k}{r \choose k}{r+k \choose k} β_{k+1}\mbox{.}

## Usage

 `1` ```pwm2lmom(pwm) ```

## Arguments

 `pwm` A PWM object created by `pwm.ub` or similar.

## Details

The probability-weighted moments (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 linearity between L-moments and PWMs means that procedures base on one are equivalent to the other.

The function can take a variety of PWM argument types in `pwm`. The function checks whether the argument is an R `list` and if so attempts to extract the β_r's from `list` names such as `BETA0`, `BETA1`, and so on. If the extraction is successful, then a list of L-moments similar to `lmom.ub` is returned. If the extraction was not successful, then an R `list` name `betas` is checked; if `betas` is found, then this vector of PWMs is used to compute the L-moments. If `pwm` is a `list` but can not be routed in the function, a `warning` is made and `NULL` is returned. If the `pwm` argument is a `vector`, then this vector of PWMs is used. to compute the L-moments are returned.

## Value

One of two R `list`s are returned. Version I is

 `L1` Arithmetic mean. `L2` L-scale—analogous to standard deviation. `LCV` coefficient of L-variation—analogous to coe. of variation. `TAU3` The third L-moment ratio or L-skew—analogous to skew. `TAU4` The fourth L-moment ratio or L-kurtosis—analogous to kurtosis. `TAU5` The fifth L-moment ratio. `L3` The third L-moment. `L4` The fourth L-moment. `L5` The fifth L-moment.

Version II is

 `lambdas` The L-moments. `ratios` The L-moment ratios. `source` Source of the L-moments “pwm2lmom”.

W.H. Asquith

## References

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`, `pwm.ub`, `pwm`, `lmom2pwm`
 ```1 2 3 4``` ```D <- c(123,34,4,654,37,78) pwm2lmom(pwm.ub(D)) pwm2lmom(pwm(D)) pwm2lmom(pwm(rnorm(100))) ```