lmomgld {lmomco} | R Documentation |
This function estimates the L-moments of the Generalized Lambda distribution given the parameters (xi, α, 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
λ_1 = xi + α (frac{1}{kappa+1} - frac{1}{h+1} ) mbox{.}
The second L-moment or L-scale in terms of the parameters and the mean is
λ_2 = xi + frac{2α}{(kappa+2)} - 2α ( frac{1}{h+1} - frac{1}{h+2} ) - xi mbox{.}
The third L-moment in terms of the parameters, the mean, and L-scale is
mbox{boldmath $Y$} = 2xi + frac{6α}{(kappa+3)} - 3(α+xi) + xi mbox{ and}
λ_3 = mbox{boldmath $Y$} + 6α (frac{2}{h+2} - frac{1}{h+3} - frac{1}{h+1}) mbox{.}
The fourth L-moment in termes of the parameters and the first three L-moments is
mbox{boldmath $Y$} = frac{-3}{h+4}(frac{2}{h+2} - frac{1}{h+3} - frac{1}{h+1}) mbox{,}
mbox{boldmath $Z$} = frac{20xi}{4} + frac{20α}{(kappa+4)} - 20 mbox{boldmath $Y$}α mbox{, and}
λ_4 = mbox{boldmath $Z$} - 5(kappa + 3(α+xi) - xi) + 6(α + 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
λ_1 = xi + α (frac{1}{kappa+1} - frac{1}{h+1} ) mbox{,}
λ_2 = α (frac{kappa}{(kappa+2)(kappa+1)} + frac{h}{(h+2)(h+1)}) mbox{,}
λ_3 = α (frac{kappa (kappa - 1)} {(kappa+3)(kappa+2)(kappa+1)} - frac{h (h - 1)} {(h+3)(h+2)(h+1)} ) mbox{, and}
λ_4 = α (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)} ) 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 >= 3 substraction is seen and for even r >= 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, ... } and h={-1, -2, -3, -4, ... } 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)
gldpara |
The parameters of the distribution. |
An R list
is returned.
L1 |
Arithmetic mean. |
L2 |
L-scale—analogous to standard deviation. |
LCV |
coefficient of L-variation—analogous to coe. of variation. |
TAU3 |
The third L-moment ratio or L-skew—analogous to skew. |
TAU4 |
The fourth L-moment ratio or L-kurtosis—analogous to kurtosis. |
TAU5 |
The fifth L-moment ratio. |
L3 |
The third L-moment. |
L4 |
The fourth L-moment. |
L5 |
The fifth L-moment. |
source |
An attribute identifying the computational source of the L-moments: “lmomgld”. |
W.H. Asquith
Derivations conducted by W.H. Asquith on February 11 and 12, 2006.
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.
pargld, \code{cdfgld}, \code{quagld}
lmomgld(vec2par(c(10,10,0.4,1.3),type='gld'))