lmomco (version 2.4.14)

parrevgum: Estimate the Parameters of the Reverse Gumbel Distribution

Description

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 2-parameter Weibull distribution. The relations between distribution parameters and L-moments are $$\alpha = \lambda^B_2/\lbrace\log(2) + \mathrm{Ei}(-2\log(1-\zeta)) - \mathrm{Ei}(-\log(1-\zeta))\rbrace$$ and $$\xi = \lambda^B_1 + \alpha\lbrace\mathrm{Ei}(-\log(1-\zeta))\rbrace\mbox{,}$$ where \(\zeta\) is the compliment of the right-tail censoring fraction of the sample or 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 \(\zeta \rightarrow 0\).

Usage

parrevgum(lmom, zeta=1, checklmom=TRUE, ...)

Value

An R

list is returned.

type

The type of distribution: revgum.

para

The parameters of the distribution.

zeta

The compliment of the right censoring fraction. Number of samples observed (noncensored) divided by the total number of samples.

source

The source of the parameters: “parrevgum”.

Arguments

lmom

An L-moment object created by lmoms through pwmRC or other L-moment type object. The user intervention of the zeta differentiates this distribution (and this function) from similar parameter estimation functions in the lmomco package.

zeta

The compliment of 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.

...

Other arguments to pass.

Author

W.H. Asquith

References

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.

See Also

lmomrevgum, cdfrevgum, pdfrevgum, quarevgum, pwm2lmom, pwmRC

Examples

# 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