lmomco (version 0.88)

lmomTLgld: Trimmed L-moments (t=1) of the Generalized Lambda Distribution

Description

This function estimates the symmetrical trimmed L-moments (TL-moments) for $t=1$ of the Generalized Lambda distribution given the parameters ($\xi$, $\alpha$, $\kappa$, and $h$) from vec2par. The TL-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 four TL-moments of the distribution are

$$\lambda^{(1)}_1 = \xi + 6\alpha \left(\frac{1}{(\kappa+3)(\kappa+2)} - \frac{1}{(h+3)(h+2)} \right) \mbox{,}$$

$$\lambda^{(1)}_2 = 6\alpha \left(\frac{\kappa}{(\kappa+4)(\kappa+3)(\kappa+2)} + \frac{h}{(h+4)(h+3)(h+2)}\right) \mbox{,}$$

$$\lambda^{(1)}_3 = \frac{20\alpha}{3} \left(\frac{\kappa (\kappa - 1)} {(\kappa+5)(\kappa+4)(\kappa+3)(\kappa+2)} - \frac{h (h - 1)} {(h+5)(h+4)(h+3)(h+2)} \right) \mbox{,}$$

$$\lambda^{(1)}_4 = \frac{15\alpha}{2} \left(\frac{\kappa (\kappa - 2)(\kappa - 1)} {(\kappa+6)(\kappa+5)(\kappa+4)(\kappa+3)(\kappa+2)} + \frac{h (h - 2)(h - 1)} {(h+6)(h+5)(h+4)(h+3)(h+2)} \right) \mbox{, and}$$ $$\lambda^{(1)}_5 = \frac{42\alpha}{5} \left(N1 - N2 \right) \mbox{,}$$

where

$$N1 = \frac{\kappa (\kappa - 3)(\kappa - 2)(\kappa - 1) } {(\kappa+7)(\kappa+6)(\kappa+5)(\kappa+4)(\kappa+3)(\kappa+2)} \mbox{ and}$$ $$N2 = \frac{h (h - 3)(h - 2)(h - 1)}{(h+7)(h+6)(h+5)(h+4)(h+3)(h+2)} \mbox{.}$$

The TL-moment ($t=1$) for $\tau^{(1)}_3$ is $$\tau^{(1)}_3 = \frac{10}{9} \left( \frac{\kappa(\kappa-1)(h+5)(h+4)(h+3)(h+2) - h(h-1)(\kappa+5)(\kappa+4)(\kappa+3)(\kappa+2)} {(\kappa+5)(h+5) \times [\kappa(h+4)(h+3)(h+2) + h(\kappa+4)(\kappa+3)(\kappa+2)] } \right) \mbox{.}$$

The TL-moment ($t=1$) for $\tau^{(1)}_4$ is $$N1 = \kappa(\kappa-2)(\kappa-1)(h+6)(h+5)(h+4)(h+3)(h+2) \mbox{,}$$ $$N2 = h(h-2)(h-1)(\kappa+6)(\kappa+5)(\kappa+4)(\kappa+3)(\kappa+2) \mbox{,}$$ $$D1 = (\kappa+6)(h+6)(\kappa+5)(h+5) \mbox{,}$$ $$D2 = [\kappa(h+4)(h+3)(h+2) + h(\kappa+4)(\kappa+3)(\kappa+2)] \mbox{, and}$$ $$\tau^{(1)}_4 = \frac{5}{4} \left( \frac{N1 + N2}{D1 \times D2} \right) \mbox{.}$$

Finally the TL-moment ($t=1$) for $\tau^{(1)}_5$ is

$$N1 = \kappa(\kappa-3)(\kappa-2)(\kappa-1)(h+7)(h+6)(h+5)(h+4)(h+3)(h+2) \mbox{,}$$ $$N2 = h(h-3)(h-2)(h-1)(\kappa+7)(\kappa+6)(\kappa+5)(\kappa+4)(\kappa+3)(\kappa+2) \mbox{,}$$ $$D1 = (\kappa+7)(h+7)(\kappa+6)(h+6)(\kappa+5)(h+5) \mbox{,}$$ $$D2 = [\kappa(h+4)(h+3)(h+2) + h(\kappa+4)(\kappa+3)(\kappa+2)] \mbox{, and}$$ $$\tau^{(1)}_5 = \frac{7}{5} \left( \frac{N1 - N2}{D1 \times D2} \right)\mbox{.}$$

By inspection the $\tau_r$ equations are not applicable for negative integer values $k={-2, -3, -4, \dots }$ and $h={-2, -3, -4, \dots }$ 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

lmomTLgld(gldpara)

Arguments

gldpara

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 = 1
• source
  An attribute identifying the computational source of the TL-moments: lmomTLgld.

source

Derivations conducted by W.H. Asquith on February 18 and 19, 2006.

References

Elamir, E.A.H., and Seheult, A.H., 2003, Trimmed L-moments: Computational statistics and data analysis, vol. 43, pp. 299-314.

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.

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.

See Also

lmomgld, pargld, cdfgld, quagld

Examples

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