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 $\beta^2\frac{\pi}{6}$.
Its standardized version is obtained with $\alpha = \frac{\sqrt{6}}{\pi}\;e$
and $\beta = \frac{\sqrt{6}}{\pi}$