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.