lmomco (version 0.6)

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.
• BETA0
  The first PWM---equal to the arithmetic mean.
• BETA1
  The second PWM.
• BETA2
  The third PWM.
• BETA3
  The fourth PWM.
• BETA4
  The fifth 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.

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., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

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)))