quarevgum: Quantile Function of the Reverse Gumbel Distribution

Description

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

$$x(F) = \xi + \alpha\log(-\log(1-F)) \mbox{,}$$

where $x(F)$ is the quantile for nonexceedance probability $F$, $\xi$ 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 Gumbel quantile function in quagum.

Usage

quarevgum(f, para)

Arguments

Nonexceedance probability ($0 \le F \le 1$).

para

The parameters from parrevgum or similar.

Value

Quantile value for nonexceedance probability $F$.

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.

Examples

Run this code

# 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
F  <- nonexceeds()
PP <- pp(D) # plotting positions of the data
plot(PP,sort(D),ylim=range(quarevgum(F,rg.pars)))
lines(F,quarevgum(F,rg.pars))
# In the plot notice how the data flat lines at the censoring level,
# but the distribution continues on.  Neat.