pwm2lmom: Probability-Weighted Moments to L-moments

In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

Probability-Weighted Moments to L-moments

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

\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{,}

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

\tau = \lambda_2/\lambda_1 \mbox{,}

\tau_3 = \lambda_3/\lambda_2 \mbox{,}

\tau_4 = \lambda_4/\lambda_2 \mbox{, and}

\tau_5 = \lambda_5/\lambda_2 \mbox{.}

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

\lambda_{r+1} = \sum^r_{k=0} (-1)^{r-k}{r \choose k}{r+k \choose k} \beta_{k+1}\mbox{.}

Usage

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 \beta_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 lists 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”.

Author(s)

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.

See Also

lmom.ub, pwm.ub, pwm, lmom2pwm

Examples

D <- c(123,34,4,654,37,78)
pwm2lmom(pwm.ub(D))
pwm2lmom(pwm(D))
pwm2lmom(pwm(rnorm(100)))

lmomco documentation built on May 29, 2024, 10:06 a.m.