pargov: Estimate the Parameters of the Govindarajulu Distribution

Description

This function estimates the parameters of the Govindarajulu distribution given the L-moments of the data in an L-moment object such as that returned by lmoms. The relations between distribution parameters and L-moments also are seen under lmomgov. The $\beta$ is estimated as $$\beta = -\frac{(4\tau_3 + 2)}{(\tau_3 - 1)}\mbox{,}$$ and $\alpha$ then $\xi$ are estimated for unknown $\xi$ as $$\alpha = \lambda_2\frac{(\beta+2)(\beta+3)}{2\beta}\mbox{, and}$$ $$\xi = \lambda_1 - \frac{2\alpha}{(\beta+2)}\mbox{,}$$ and $\alpha$ is estimated for known $\xi$ as $$\alpha = (\lambda_1 - \xi)\frac{(\beta + 2)}{2}\mbox{.}$$ The shape preservation for this distribution is an ad hoc decision. It could be that for given $\xi$, that solutions could fall back to estimating $\xi$ and $\alpha$ from $\lambda_1$ and $\lambda_2$ only. Such as solution would rely on $\tau_2 = \lambda_2/\lambda_1$ with $\beta$ estimated as $$\beta = \frac{3\tau_2}{(1-\tau_2)}\mbox{, and}$$ $$\alpha = \lambda_1\frac{(\beta+2)}{2}\mbox{,}$$ but such a practice yields remarkable changes in shape for this distribution even if the provided $\xi$ precisely matches that from a previous parameter estimation for which the $\xi$ was treated as unknown.

Usage

pargov(lmom, xi=NULL, checklmom=TRUE, ...)

Value

An R

list is returned.

type: The type of distribution: gov.
para: The parameters of the distribution.
source: The source of the parameters: “pargov”.

Arguments

lmom: An L-moment object created by lmoms or vec2lmom.
xi: An optional lower limit of the distribution. If not NULL, the $B$ is still uniquely determined by $\tau_3$, the $\alpha$ is adjusted so that the given lower bounds is honored. It is generally accepted to let the distribution fitting process determine its own lower bounds so xi=NULL should suffice in many circumstances.
checklmom: Should the lmom be checked for validity using the are.lmom.valid function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the $\tau_4$ and $\tau_3$ inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check.
...: Other arguments to pass.

Author

W.H. Asquith

References

Gilchrist, W.G., 2000, Statistical modelling with quantile functions: Chapman and Hall/CRC, Boca Raton.

Nair, N.U., Sankaran, P.G., Balakrishnan, N., 2013, Quantile-based reliability analysis: Springer, New York.

Nair, N.U., Sankaran, P.G., and Vineshkumar, B., 2012, The Govindarajulu distribution---Some Properties and applications: Communications in Statistics, Theory and Methods, v. 41, no. 24, pp. 4391--4406.

Examples

Run this code

lmr <- lmoms(rnorm(20))
pargov(lmr)

lmr <- vec2lmom(c(1391.8, 215.68, 0.01655, 0.09628))
pargov(lmr)$para             # see below
#         xi       alpha        beta 
# 868.148125 1073.740595    2.100971 
pargov(lmr, xi=868)$para     # see below
#         xi       alpha        beta 
# 868.000000 1074.044324    2.100971 
pargov(lmr, xi=100)$para     # see below
#         xi       alpha        beta 
# 100.000000 2648.817215    2.100971

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