vec2par. The L-moments in terms of the parameters are complicated; however, there are analytical solutions. There are no simple expressions of the parameters in terms of the L-moments. The first L-moment or the mean of the distribution is$$\lambda_1 = \xi + \alpha \left(\frac{1}{\kappa+1} - \frac{1}{h+1} \right) \mbox{.}$$
The second L-moment or L-scale in terms of the parameters and the mean is
$$\lambda_2 = \xi + \frac{2\alpha}{(\kappa+2)} - 2\alpha \left( \frac{1}{h+1} - \frac{1}{h+2} \right) - \xi \mbox{.}$$
The third L-moment in terms of the parameters, the mean, and L-scale is
$$\mbox{\boldmath $Y$} = 2\xi + \frac{6\alpha}{(\kappa+3)} - 3(\alpha+\xi) + \xi \mbox{ and}$$ $$\lambda_3 = \mbox{\boldmath $Y$} + 6\alpha \left(\frac{2}{h+2} - \frac{1}{h+3} - \frac{1}{h+1}\right) \mbox{.}$$
The fourth L-moment in termes of the parameters and the first three L-moments is
$$\mbox{\boldmath $Y$} = \frac{-3}{h+4}\left(\frac{2}{h+2} - \frac{1}{h+3} - \frac{1}{h+1}\right) \mbox{,}$$ $$\mbox{\boldmath $Z$} = \frac{20\xi}{4} + \frac{20\alpha}{(\kappa+4)} - 20 \mbox{\boldmath $Y$}\alpha \mbox{, and}$$ $$\lambda_4 = \mbox{\boldmath $Z$} - 5(\kappa + 3(\alpha+\xi) - \xi) + 6(\alpha + \xi) - \xi \mbox{.}$$
It is conventional to express L-moments in terms of only the parameters and not the other L-moments. Lengthy algebra and further manipulation yields such a system of equations. The L-moments of the distribution are
$$\lambda_1 = \xi + \alpha \left(\frac{1}{\kappa+1} - \frac{1}{h+1} \right) \mbox{,}$$
$$\lambda_2 = \alpha \left(\frac{\kappa}{(\kappa+2)(\kappa+1)} + \frac{h}{(h+2)(h+1)}\right) \mbox{,}$$
$$\lambda_3 = \alpha \left(\frac{\kappa (\kappa - 1)} {(\kappa+3)(\kappa+2)(\kappa+1)} - \frac{h (h - 1)} {(h+3)(h+2)(h+1)} \right) \mbox{, and}$$
$$\lambda_4 = \alpha \left(\frac{\kappa (\kappa - 2)(\kappa - 1)} {(\kappa+4)(\kappa+3)(\kappa+2)(\kappa+1)} + \frac{h (h - 2)(h - 1)} {(h+4)(h+3)(h+2)(h+1)} \right) \mbox{.}$$
The L-moment ratios are $$\tau_3 = \frac{\kappa(\kappa-1)(h+3)(h+2)(h+1) - h(h-1)(\kappa+3)(\kappa+2)(\kappa+1)} {(\kappa+3)(h+3) \times [\kappa(h+2)(h+1) + h(\kappa+2)(\kappa+1)] } \mbox{ and}$$
$$\tau_4 = \frac{\kappa(\kappa-2)(\kappa-1)(h+4)(h+3)(h+2)(h+1) + h(h-2)(h-1)(\kappa+4)(\kappa+3)(\kappa+2)(\kappa+1)} {(\kappa+4)(h+4)(\kappa+3)(h+3) \times [\kappa(h+2)(h+1) + h(\kappa+2)(\kappa+1)] } \mbox{.}$$
The pattern being established through symmetry, even higher L-moment ratios are readily obtained. Note the alternating substraction and addition of the two terms in the numerator of the L-moment ratios ($\tau_r$). For odd $r \ge 3$ substraction is seen and for even $r \ge 3$ addition is seen. For example, the fifth L-moment ratio is
$$N1 = \kappa(\kappa-3)(\kappa-2)(\kappa-1)(h+5)(h+4)(h+3)(h+2)(h+1) \mbox{,}$$ $$N2 = h(h-3)(h-2)(h-1)(\kappa+5)(\kappa+4)(\kappa+3)(\kappa+2)(\kappa+1) \mbox{,}$$ $$D1 = (\kappa+5)(h+5)(\kappa+4)(h+4)(\kappa+3)(h+3) \mbox{,}$$ $$D2 = [\kappa(h+2)(h+1) + h(\kappa+2)(\kappa+1)] \mbox{, and}$$ $$\tau_5 = \frac{N1 - N2}{D1 \times D2} \mbox{.}$$
By inspection the $\tau_r$ equations are not applicable for negative integer values $k={-1, -2, -3, -4, \dots }$ and $h={-1, -2, -3, -4, \dots }$ as division by zero will result. There are additional, but difficult to formulate, restrictions on the parameters both to define a valid Generalized Lambda distribution as well as valid L-moments. Verification of the parameters is conducted through are.pargld.valid, and verification of the L-moment validity is conducted through are.lmom.valid.
lmomgld(gldpara)list is returned.Karvanen, J., Eriksson, J., and Koivunen, V., 2002, Adaptive score functions for maximum likelihood ICA: Journal of VLSI Signal Processing, vol. 32, p. 82--92.
Karian, Z.A., and Dudewicz, E.J., 2000, Fitting statistical distibutions---The generalized lambda distribution and generalized bootstrap methods: CRC Press, Boca Raton, FL, 438 p.
pargld, cdfgld, quagldlmomgld(vec2par(c(10,10,0.4,1.3),type='gld'))Run the code above in your browser using DataLab