lmomco (version 0.88)

lmom2pwm: L-moments to Probability-Weighted Moments

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 relation between L-moments and PWMs is shown with pwm2lmom.

Usage

lmom2pwm(lmom)

Arguments

lmom

An L-moment object created by lmom.ub or similar.

Value

• An R list is returned.
• betas
  The PWMs. Note that convention is the have a $\beta_0$, but this is placed in the first index i=1 of the betas vector.
• source
  Source of the PWMs: pwm

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 here for theoretical completeness and are not intended for use with the majority of the other functions implementing the various probability distributions. The relation between L-moments ($\lambda_r$) and PWMs ($\beta_{r-1}$) for $1 \le r \le 5$ order is

$$\lambda_1 = \beta_0 \mbox{,}$$ $$\lambda_2 = 2\beta_1 - \beta_0 \mbox{,}$$ $$\lambda_3 = 6\beta_2 - 6\beta_1 + \beta_0 \mbox{,}$$ $$\lambda_4 = 20\beta_3 - 30\beta_2 + 12\beta_1 - \beta_0\mbox{, and}$$

$$\lambda_5 = 70\beta_4 - 140\beta_3 + 90\beta_2 - 20\beta_1 + \beta_0\mbox{.}$$

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.

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, vol. 15, p. 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, vol. 52, p. 105--124.

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press.

See Also

lmom.ub, pwm.ub, pwm2lmom

Examples

pwm <- lmom2pwm(lmom.ub(c(123,34,4,654,37,78)))

lmom2pwm(lmom.ub(rnorm(100)))