lmomco (version 1.7.3)

pdfrevgum: Probability Density Function of the Reverse Gumbel Distribution

Description

This function computes the probability density of the Reverse Gumbel distribution given parameters ($\xi$ and $\alpha$) of the distribution computed by parrevgum. The probability density function of the distribution is

$$f(x) = \alpha^{-1} e^y [e^{-e^{y}}] \mbox{,}$$

where

$$y = \frac{x - \xi}{\alpha} \mbox{,}$$

where $f(x)$ is the probability density for quantile $x$, $\xi$ is a location parameter, and $\alpha$ is a scale parameter. Notice that the function has some sign differences and uses the complement of $f$ compared to the probability density function of the Gumbel distribution in pdfgum.

Usage

pdfrevgum(x, para)

Arguments

x

A real value.

para

The parameters from parrevgum or similar.

Value

• Probability density ($f$) for $x$.

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

cdfrevgum, quarevgum, parrevgum

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 = -0.516 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
# Now one can continue one with a plotting example.
F  <- nonexceeds()
PP <- pp(D) # plotting positions of the data
D  <- sort(D)
plot(D,PP)
lines(D,cdfrevgum(D,rg.pars))
# Now finally do the PDF
F <- seq(0.01,0.99,by=.01)
x <- quarevgum(F,rg.pars)
plot(x,pdfrevgum(x,rg.pars),type='l')