17.9 Gumbel distribution

17.9.1 Probability density function

If \(\mu \in \mathbb{R}\) and \(\beta \in \mathbb{R}^+\), then for \(y \in \mathbb{R}\), \[ \text{Gumbel}(y|\mu,\beta) = \frac{1}{\beta} \ \exp\left(-\frac{y-\mu}{\beta}-\exp\left(-\frac{y-\mu}{\beta}\right)\right) . \]

17.9.2 Sampling statement

y ~ gumbel(mu, beta)

Increment target log probability density with gumbel_lupdf(y | mu, beta).
Available since 2.0

17.9.3 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
Available since 2.12

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
Available since 2.25

real gumbel_cdf(reals y, reals mu, reals beta)
The gumbel cumulative distribution function of y given location mu and scale beta
Available since 2.0

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
Available since 2.12

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
Available since 2.12

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.
Available since 2.18