lmomco (version 1.7.3)

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 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 $\zeta \rightarrow 0$.

Usage

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

Arguments

lmom

A L-moment object created by pwm2lmom through pwmRC or other L-moment type object. The user intervention of the zeta differentiates this distribu

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 $\tau_3$ inequ

Value

• 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.

References

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.

See Also

pwm2lmom, pwmRC, cdfrevgum, quarevgum

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