This function estimates the parameters of the Reverse Gumbel distribution given
the type-B L-moments of the data in an L-moment object such as that returned by
pwmRC using pwm2lmom. This distribution is important in the analysis of censored data. It is the distribution of a logarithmically transformed two-parameter Weibull distribution. The relation between distribution parameters and L-moments
is$$\alpha = \lambda^B_2/\lbrace\log(2) + \mathrm{Ei}(-2\log(1-\zeta)) - \mathrm{Ei}(-\log(1-\zeta))\rbrace\mbox{\ and}$$
$$\xi = \lambda^B_1 + \alpha\lbrace\mathrm{Ei}(-\log(1-\zeta))\rbrace\mbox{,}$$
where $\zeta$ is the right-tail censoring fraction of the sample o the nonexceedance probability of the right-tail censoring threshold, and $\mathrm{Ei}(x)$ is the exponential integral defined as
$$\mathrm{Ei}(X) = \int_X^{\infty} x^{-1}e^{-x}\mathrm{d}x \mbox{,}$$
where $\mathrm{Ei}(-\log(1-\zeta)) \rightarrow 0$ as $\zeta \rightarrow 1$ and $\mathrm{Ei}(-\log(1-\zeta))$ can not be evaluated as $\zeta \rightarrow 0$.
parrevgum(lmom,zeta=1,checklmom=TRUE)
- lmom
{A L-moment object created by pwm2lmom
through pwmRC or other L-moment type object. The user intervention of the zeta differentiates this distribution and hence this function from similar parameter estimation functions in the lmomco package.}
- zeta
{The right censoring fraction. Number of samples observed (noncensored) divided by the total number of samples.}
- 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.}
An R list is returned.
- type
{The type of distribution: revgum.}
- para
{The parameters of the distribution.}
- zeta
{The right censoring fraction. Number of samples observed (noncensored) divided by the total number of samples.}
- source
{The source of the parameters: parrevgum.}
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.
Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data,
in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan,
chapter 29, CRC Press, Boca Raton, Fla., pp. 546--560.
[object Object]
pwm2lmom, pwmRC,
cdfrevgum, quarevgum
# See p. 553 of Hosking (1995)
# Data listed in Hosking (1995, table 29.3, p. 553)
D <- c(-2.982, -2.849, -2.546, -2.350, -1.983, -1.492, -1.443,
-1.394, -1.386, -1.269, -1.195, -1.174, -0.854, -0.620,
-0.576, -0.548, -0.247, -0.195, -0.056, -0.013, 0.006,
0.033, 0.037, 0.046, 0.084, 0.221, 0.245, 0.296)
D <- c(D,rep(.2960001,40-28)) # 28 values, but Hosking mentions 40 values in total
z <- pwmRC(D,threshold=.2960001)
str(z)
# Hosking reports B-type L-moments for this sample are
# lamB1 = -.516 and lamB2 = 0.523
btypelmoms <- pwm2lmom(z$Bbetas)
# My version of R reports lamB1 = -0.5162 and lamB2 = 0.5218
str(btypelmoms)
rg.pars <- parrevgum(btypelmoms,z$zeta)
str(rg.pars)
# Hosking reports xi = 0.1636 and alpha = 0.9252 for the sample
# My version of R reports xi = 0.1635 and alpha = 0.9254
distribution