lmomgld: L-moments of the Generalized Lambda Distribution

In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

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 \lambda_1, second element is \lambda_2, and so on.
ratios	Vector of the L-moment ratios. Second element is \tau, third element is \tau_3 and so on.
trim	Level of symmetrical trimming used in the computation, which is 0.
leftrim	Level of left-tail trimming used in the computation, which is NULL.
rightrim	Level of right-tail trimming used in the computation, which is NULL.
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)

wasquith/lmomco documentation built on Nov. 13, 2024, 4:53 p.m.