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 $$\lambda_{r+1} = \sum^r_{k=0} (-1)^{r-k}{r \choose k}{r+k \choose k} \beta_{k+1}\mbox{.}$$
pwm2lmom(pwm)One of two R
lists are returned. Version I is
Arithmetic mean.
L-scale---analogous to standard deviation.
coefficient of L-variation---analogous to coe. of variation.
The third L-moment ratio or L-skew---analogous to skew.
The fourth L-moment ratio or L-kurtosis---analogous to kurtosis.
The fifth L-moment ratio.
The third L-moment.
The fourth L-moment.
The fifth L-moment.
Version II is
The L-moments.
The L-moment ratios.
Source of the L-moments “pwm2lmom”.
A PWM object created by pwm.ub or similar.
W.H. Asquith
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.
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
D <- c(123,34,4,654,37,78)
pwm2lmom(pwm.ub(D))
pwm2lmom(pwm(D))
pwm2lmom(pwm(rnorm(100)))
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