extreme value:
Density of the Extreme Value Distribution of a Minimum.
Description
Density function of the extreme value distribution of a minimum
with location $alpha$ and scale $beta$
and the density of the standardized version (with zero mean and unit variance).
Usage
dextreme(x, alpha=0, beta=1)
dstextreme(x)
Arguments
x
Vector of quantiles.
alpha
Vector of location parameters.
beta
Vector of scale parameters.
Value
The value of the density.
Details
Extreme value distribution of a minimum with the location $alpha$
and the scale $beta$ has a density
$$f(x) = \frac{1}{\beta}\exp\left[\frac{x-\alpha}{\beta}-\exp\left(\frac{x-\alpha}{\beta}\right)\right]$$
the mean equal to $alpha - beta*e$, where $e$ is approximately
$0.5772$ and the variance equal to $beta^2 pi^2/6$.
Its standardized version is obtained with $alpha = (sqrt(6)/pi)*e$
and $beta = (sqrt(6)/pi).$