DAMisc (version 1.7.2)

pgumbel: PDF of the Gumbel Distribution

Description

Returns the PDF of the Gumbel distribution.

Usage

pgumbel(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

q

Vector of quantiles

location

The location parameter, $mu$. This is not the mean of the Gumbel distribution.

scale

The scale parameter $sigma$.

lower.tail

Logical, whether lower, if TRUE or upper, if FALSE, tail probabilities should be returned.

log.p

Logical, if TRUE probabilities are given in their log.

Value

A vector of probabilities

Details

This code and the details of the help file were taken from the VGAM package.

The Gumbel distribution is a special case of the generalized extreme value (GEV) distribution where the shape parameter $\xi$ = 0. The latter has 3 parameters, so the Gumbel distribution has two. The Gumbel distribution function is $$G(y)=\exp (-\exp [-\frac{y-\mu }{\sigma }])$$ where $-\mathrm{\infty }<y<\mathrm{\infty }$, $-\mathrm{\infty }<\mu <\mathrm{\infty }$and $\sigma >0$. Its mean is $$\mu -\sigma \ast \gamma$$ and its variance is $${\sigma }^2\ast {\pi }^2/6$$ where $\gamma$ is Euler's constant (which can be obtained as -digamma(1)).