16.9 Gumbel distribution
16.9.1 Probability density function
If \(\mu \in \mathbb{R}\) and \(\beta \in \mathbb{R}^+\), then for \(y \in \mathbb{R}\), \[ \text{Gumbel}(y|\mu,\beta) = \frac{1}{\beta} \ \exp\left(-\frac{y-\mu}{\beta}-\exp\left(-\frac{y-\mu}{\beta}\right)\right) . \]
16.9.2 Sampling statement
y ~
gumbel
(mu, beta)
Increment target log probability density with gumbel_lupdf(y | mu, beta)
.
16.9.3 Stan functions
real
gumbel_lpdf
(reals y | reals mu, reals beta)
The log of the gumbel density of y given location mu and scale beta
real
gumbel_lupdf
(reals y | reals mu, reals beta)
The log of the gumbel density of y given location mu and scale beta
dropping constant additive terms
real
gumbel_cdf
(reals y, reals mu, reals beta)
The gumbel cumulative distribution function of y given location mu and
scale beta
real
gumbel_lcdf
(reals y | reals mu, reals beta)
The log of the gumbel cumulative distribution function of y given
location mu and scale beta
real
gumbel_lccdf
(reals y | reals mu, reals beta)
The log of the gumbel complementary cumulative distribution function
of y given location mu and scale beta
R
gumbel_rng
(reals mu, reals beta)
Generate a gumbel variate with location mu and scale beta; may only be
used in transformed data and generated quantities blocks. For a description
of argument and return types, see section
vectorized PRNG functions.