pdfrevgum {lmomco}R Documentation

Probability Density Function of the Reverse Gumbel Distribution

Description

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

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

where

y = frac{x - xi}{α} mbox{,}

where f(x) is the probability density for quantile x, xi is a location parameter, and α 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.

Author(s)

W.H. Asquith

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
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')

[Package lmomco version 0.96.3 Index]