Cumulative Distribution Function of the Reverse Gumbel Distribution

Description

This function computes the cumulative probability or nonexceedance probability of the Reverse Gumbel distribution given parameters (ξ and α) computed by parrevgum. The cumulative distribution function is

F(x) = 1 - \mathrm{exp}(-\mathrm{exp}(Y)) \mbox{,}

where

Y = -\frac{x - ξ}{α} \mbox{,}

where F(x) is the nonexceedance probability for quantile x, ξ is a location parameter, and α is a scale parameter.

Usage

1
cdfrevgum(x, para)

Arguments

x

A real value vector.

para

The parameters from parrevgum or vec2par.

Value

Nonexceedance probability (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, v. 52, pp. 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

pdfrevgum, quarevgum, lmomrevgum, parrevgum

Examples

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# 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))

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