lmomco (version 1.7.3)

lmomgld: L-moments of the Generalized Lambda Distribution

Description

This function estimates the L-moments of the Generalized Lambda distribution given the parameters ($\xi$, $\alpha$, $\kappa$, and $h$) from 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

$${\lambda }_1=\xi +\alpha (\frac{1}{\kappa +1}-\frac{1}{h+1})\text{.}$$

The second L-moment or L-scale in terms of the parameters and the mean is

$${\lambda }_2=\xi +\frac{2\alpha }{(\kappa +2)}-2\alpha (\frac{1}{h+1}-\frac{1}{h+2})-\xi \text{.}$$

The third L-moment in terms of the parameters, the mean, and L-scale is

$$\text{boldmath }Y=2\xi +\frac{6\alpha }{(\kappa +3)}-3(\alpha +\xi )+\xi \text{ and}$$ $${\lambda }_3=\text{boldmath }Y+6\alpha (\frac{2}{h+2}-\frac{1}{h+3}-\frac{1}{h+1})\text{.}$$

The fourth L-moment in termes of the parameters and the first three L-moments is

$$\text{boldmath }Y=\frac{-3}{h+4}(\frac{2}{h+2}-\frac{1}{h+3}-\frac{1}{h+1})\text{,}$$ $$\text{boldmath }Z=\frac{20\xi }{4}+\frac{20\alpha }{(\kappa +4)}-20\text{boldmath }Y\alpha \text{, and}$$ $${\lambda }_4=\text{boldmath }Z-5(\kappa +3(\alpha +\xi )-\xi )+6(\alpha +\xi )-\xi \text{.}$$

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

$${\lambda }_1=\xi +\alpha (\frac{1}{\kappa +1}-\frac{1}{h+1})\text{,}$$

$${\lambda }_2=\alpha (\frac{\kappa }{(\kappa +2)(\kappa +1)}+\frac{h}{(h+2)(h+1)})\text{,}$$

$${\lambda }_3=\alpha (\frac{\kappa (\kappa -1)}{(\kappa +3)(\kappa +2)(\kappa +1)}-\frac{h(h-1)}{(h+3)(h+2)(h+1)})\text{, and}$$

$${\lambda }_4=\alpha (\frac{\kappa (\kappa -2)(\kappa -1)}{(\kappa +4)(\kappa +3)(\kappa +2)(\kappa +1)}+\frac{h(h-2)(h-1)}{(h+4)(h+3)(h+2)(h+1)})\text{.}$$

The L-moment ratios are

$${\tau }_3=\frac{\kappa (\kappa -1)(h+3)(h+2)(h+1)-h(h-1)(\kappa +3)(\kappa +2)(\kappa +1)}{(\kappa +3)(h+3)\times [\kappa (h+2)(h+1)+h(\kappa +2)(\kappa +1)]}\text{ and}$$

$${\tau }_4=\frac{\kappa (\kappa -2)(\kappa -1)(h+4)(h+3)(h+2)(h+1)+h(h-2)(h-1)(\kappa +4)(\kappa +3)(\kappa +2)(\kappa +1)}{(\kappa +4)(h+4)(\kappa +3)(h+3)\times [\kappa (h+2)(h+1)+h(\kappa +2)(\kappa +1)]}\text{.}$$

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

$$N1=\kappa (\kappa -3)(\kappa -2)(\kappa -1)(h+5)(h+4)(h+3)(h+2)(h+1)\text{,}$$ $$N2=h(h-3)(h-2)(h-1)(\kappa +5)(\kappa +4)(\kappa +3)(\kappa +2)(\kappa +1)\text{,}$$ $$D1=(\kappa +5)(h+5)(\kappa +4)(h+4)(\kappa +3)(h+3)\text{,}$$ $$D2=[\kappa (h+2)(h+1)+h(\kappa +2)(\kappa +1)]\text{, and}$$ $${\tau }_5=\frac{N1-N2}{D1\times D2}\text{.}$$

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.

Usage

lmomgld(para)

Arguments

para

The parameters of the distribution.

Value

• An R list is returned.
• lambdas
  Vector of the TL-moments. First element is $\lambda^{(1)}_1$, second element is $\lambda^{(1)}_2$, and so on.
• ratios
  Vector of the TL-moment ratios. Second element is $\tau^{(1)}$, third element is $\tau^{(1)}_3$ and so on.
• trim
  Trim level = 0
• leftrim
  Left trimming level = 0
• rightrim
  Right trimming level = 0
• source
  An attribute identifying the computational source of the TL-moments: lmomgld.

source

Derivations conducted by W.H. Asquith on February 11 and 12, 2006.

References

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.

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.

See Also

pargld, cdfgld, quagld

Examples

lmomgld(vec2par(c(10,10,0.4,1.3),type='gld'))

PARgld <- vec2par(c(0,1,1,.5), type="gld")
theoTLmoms(PARgld, nmom=6)
lmomgld(PARgld)