# theopwms: The Theoretical Probability-Weighted Moments using... In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

Compute the theoretrical probability-weighted moments (PWMs) for a distribution. A theoretrical PWM in integral form is

β_r = \int^1_0 x(F)\,F^r\,\mathrm{d}F \mbox{,}

where x(F) is the quantile function of the random variable X for nonexceedance probability F and r represents the order of the PWM. This function loops across the above equation for each nmom set in the argument list. The function x(F) is computed through the par2qua function. The distribution type is determined using the type attribute of the para argument, which is a parameter object of lmomco (see vec2par).

## Usage

 1 theopwms(para, nmom=5, verbose=FALSE) 

## Arguments

 para A distribution parameter object such as that by lmom2par or vec2par. nmom The number of moments to compute. Default is 5. verbose Toggle verbose output. Because the R function integrate is used to perform the numerical integration, it might be useful to see selected messages regarding the numerical integration.

## 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 An attribute identifying the computational source of the probability-weighted moments: “theopwms”.

W.H. Asquith

## References

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, p. 105–124.

theoLmoms, pwm, pwm2lmom
 1 2 3 para <- vec2par(c(0,1),type='nor') # standard normal the.pwms <- theopwms(para) # compute PWMs str(the.pwms)