| lmomgld {lmomco} | R Documentation |
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 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
Y = 2\xi + \frac{6\alpha}{(\kappa+3)} -
3(\alpha+\xi) + \xi \mbox{, and}
\lambda_3 = 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
Y = \frac{-3}{h+4}\left(\frac{2}{h+2} -
\frac{1}{h+3} -
\frac{1}{h+1}\right) \mbox{,}
Z = \frac{20\xi}{4} + \frac{20\alpha}{(\kappa+4)} -
20 Y\alpha \mbox{, and}
\lambda_4 = 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 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.
Usage
lmomgld(para)
Arguments
para |
The parameters of the distribution. |
Value
An R list is returned.
lambdas |
Vector of the L-moments. First element is
|
ratios |
Vector of the L-moment ratios. Second element is
|
trim |
Level of symmetrical trimming used in the computation, which is |
leftrim |
Level of left-tail trimming used in the computation, which is |
rightrim |
Level of right-tail trimming used in the computation, which is |
source |
An attribute identifying the computational source of the L-moments: “lmomgld”. |
Author(s)
W.H. Asquith
Source
Derivations conducted by W.H. Asquith on February 11 and 12, 2006.
References
Asquith, W.H., 2007, L-moments and TL-moments of the generalized lambda distribution: Computational Statistics and Data Analysis, v. 51, no. 9, pp. 4484–4496.
Karvanen, J., Eriksson, J., and Koivunen, V., 2002, Adaptive score functions for maximum likelihood ICA: Journal of VLSI Signal Processing, v. 32, pp. 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, pdfgld, quagld
Examples
## Not run:
lmomgld(vec2par(c(10,10,0.4,1.3),type='gld'))
## End(Not run)
## Not run:
PARgld <- vec2par(c(0,1,1,.5), type="gld")
theoTLmoms(PARgld, nmom=6)
lmomgld(PARgld)
## End(Not run)