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, paracheck=TRUE)

Arguments

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

para

The parameters from parrevgum or similar.

paracheck

A logical controlling whether the parameters and checked for validity. Overriding of this check might be extremely important and needed for use of the distribution quantile function in the context of TL-moments with nonzero trimming.

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.