reliaR (version 0.01)

Gumbel: The Gumbel distribution

Description

Density, distribution function, quantile function and random generation for the Gumbel distribution with location parameter mu and scale parameter sigma.

Usage

dgumbel(x, mu, sigma, log = FALSE)
pgumbel(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qgumbel(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rgumbel(n, mu, sigma)

Arguments

x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

mu

location parameter.

sigma

scale parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $P[X \le x]$ otherwise, $P[X > x]$.

Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, and rgumbel generates random deviates.

Details

The Gumbel distribution has density $$f(x)=\frac{1}{\sigma }\exp \{-\left(\frac{x-\mu }{\sigma }\right)\}\exp [-\exp \{-\left(\frac{x-\mu }{\sigma }\right)\}];-\mathrm{\infty }<x<\mathrm{\infty },\sigma >0.$$ where $\mu$ and $\sigma$ are the shape and scale parameters, respectively.

References

Marshall, A. W., Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

.Random.seed about random number; sgumbel for Gumbel survival / hazard etc. functions

Examples

## Load data sets
data(dataset2)
## Maximum Likelihood(ML) Estimates of mu & sigma for the data(dataset2)
## Estimates of mu & sigma using 'maxLik' package
## mu.est = 212.157, sigma.est = 151.768

dgumbel(dataset2, 212.157, 151.768, log = FALSE)
pgumbel(dataset2, 212.157, 151.768, lower.tail = TRUE, log.p = FALSE)
qgumbel(0.25, 212.157, 151.768, lower.tail=TRUE, log.p = FALSE)
rgumbel(30, 212.157, 151.768)