$$\lambda_1 = \beta_0 \mbox{,}$$ $$\lambda_2 = 2\beta_1 - \beta_0 \mbox{,}$$ $$\lambda_3 = 6\beta_2 - 6\beta_1 + \beta_0 \mbox{,}$$ $$\lambda_4 = 20\beta_3 - 30\beta_2 + 12\beta_1 - \beta_0 \mbox{,}$$ $$\lambda_5 = 70\beta_4 - 140\beta_3 + 90\beta_2 - 20\beta_1 + \beta_0 \mbox{,}$$ $$\tau = \lambda_2/\lambda_1 \mbox{,}$$ $$\tau_3 = \lambda_3/\lambda_2 \mbox{,}$$ $$\tau_4 = \lambda_4/\lambda_2 \mbox{, and}$$ $$\tau_5 = \lambda_5/\lambda_2 \mbox{.}$$
The general expression and the expression used for computation if the argument is a vector of PWMs is
$$\lambda_{r+1} = \sum^r_{k=0} (-1)^{r-k}{r \choose k}{r+k \choose k} \beta_{k+1}$$
pwm2lmom(pwm)pwm.ub or similar.list are returned.The function can take a variety of PWM argument types in pwm. The function checks whether the argument is a 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 a 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 warnings is made and NULL returned. If the pwm argument is a vector, then this vector of PWMs is used to compute the L-moments are returned.
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.
lmom.ub, pwm.ub, pwm, lmom2pwmD <- c(123,34,4,654,37,78)
pwm2lmom(pwm.ub(D))
pwm2lmom(pwm(D))
pwm2lmom(pwm(rnorm(100)))Run the code above in your browser using DataLab