lmom.ub or a trimmed L-moment object (TLmoms for t=1. The relation between distribution parameters and L-moments
is seen under lmomgld. There are no simple expressions for the parameters in terms of the
L-moments. This function is considered HIGHLY EXPERIMENTAL and general details of the algorithm are provided below. Further, consider that multiple parameter solutions are possible with the Generalized Lambda so some expertise in the distribution and other aspects are needed.pargld(lmom,result='best',verbose=FALSE,extract=0,initkh=NULL)best, then the minimum error solution is returned. If dataframe, then data.frame is returned with sequence of valid solutions sorted in ascending error order.verbose=FALSE.result=dataframe and extract greater than zero, then the extract=n returns the nth element of the data.frame as if that element was the best solution.list is returned if result='best'.gld.data.frame is returned if result='dataframe', which is sorted by ascending error.optim (the R function) efforts to perform a least sum-of-square errors on the following objective function.$$(\hat{\tau}_3 - \tilde{\tau}_3)^2 + (\hat{\tau}_4 - \tilde{\tau}_4)^2 \mbox{, }$$
where $\hat{\tau}_r$ is the L-moment ratio of the data, $\tilde{\tau}_r$ is the estimated value of the L-moment ratio for the fitted distribution $\kappa$ and $h$ and $\tau_r$ is the actual value of the L-moment ratio.
For each optimization a check on the validity of the parameters so produced is made--are the parameters consistent with the Generalized Lambda distribution and a second check is made on the validity of $\tau_3$ and $\tau_4$. If both validity checks return TRUE then the optimization is retained if its sum-of-square error is less than the previous optimum value. It is possible for a given solution to be found outside the starting region of the initial guesses. The surface generated by the $\tau_3$ and $\tau_4$ equations seen in lmomgld is complex--different initial guesses within a given region can yield what appear to be radically different $\kappa$ and $h$. Users are encouraged to verbose argument). A quick double check on the L-moments from the solved parameters using lmomgld is encouraged as well. Karvanen and others (2002, eq. 25) provide an equation expressing $\kappa$ and $h$ as equal (a symmetrical Generalized Lambda distribution) in terms of $\tau_4$ and suggest that the equation be used to determine initial values for the parameters. This equation is used on an experimental basis for the final optimization attempt by this function.
Karian, Z.A., and Dudewicz, E.J., 2000, Fitting statistical distributions---The generalized lambda distribution and generalized bootstrap methods: CRC Press, Boca Raton, FL, 438 p.
lmom.ub, lmomgld, cdfgld, quagld, parTLgldlmr1 <- lmom.ub(rnorm(200))
P <- pargld(lmr1)
lmr2 <- lmomgld(P)Run the code above in your browser using DataLab