```1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173```
``````// Copyright 2021 Developers of the Rand project.
//
// option. This file may not be copied, modified, or distributed
// except according to those terms.

//! The Gumbel distribution `Gumbel(μ, β)`.

use crate::{Distribution, OpenClosed01};
use core::fmt;
use num_traits::Float;
use rand::Rng;

/// The [Gumbel distribution](https://en.wikipedia.org/wiki/Gumbel_distribution) `Gumbel(μ, β)`.
///
/// The Gumbel distribution is a continuous probability distribution
/// with location parameter `μ` (`mu`) and scale parameter `β` (`beta`).
/// It is used to model the distribution of the maximum (or minimum)
/// of a number of samples of various distributions.
///
/// # Density function
///
/// `f(x) = exp(-(z + exp(-z))) / β`, where `z = (x - μ) / β`.
///
/// # Plot
///
/// The following plot illustrates the Gumbel distribution with various values of `μ` and `β`.
/// Note how the location parameter `μ` shifts the distribution along the x-axis,
/// and the scale parameter `β` changes the density around `μ`.
/// Note also the asymptotic behavior of the distribution towards the right.
///
/// ![Gumbel distribution](https://raw.githubusercontent.com/rust-random/charts/main/charts/gumbel.svg)
///
/// # Example
/// ```
/// use rand::prelude::*;
/// use rand_distr::Gumbel;
///
/// let val: f64 = thread_rng().sample(Gumbel::new(0.0, 1.0).unwrap());
/// println!("{}", val);
/// ```
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
pub struct Gumbel<F>
where
F: Float,
OpenClosed01: Distribution<F>,
{
location: F,
scale: F,
}

/// Error type returned from [`Gumbel::new`].
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum Error {
/// location is infinite or NaN
LocationNotFinite,
/// scale is not finite positive number
ScaleNotPositive,
}

impl fmt::Display for Error {
fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
f.write_str(match self {
Error::ScaleNotPositive => "scale is not positive and finite in Gumbel distribution",
Error::LocationNotFinite => "location is not finite in Gumbel distribution",
})
}
}

#[cfg(feature = "std")]
impl std::error::Error for Error {}

impl<F> Gumbel<F>
where
F: Float,
OpenClosed01: Distribution<F>,
{
/// Construct a new `Gumbel` distribution with given `location` and `scale`.
pub fn new(location: F, scale: F) -> Result<Gumbel<F>, Error> {
if scale <= F::zero() || scale.is_infinite() || scale.is_nan() {
return Err(Error::ScaleNotPositive);
}
if location.is_infinite() || location.is_nan() {
return Err(Error::LocationNotFinite);
}
Ok(Gumbel { location, scale })
}
}

impl<F> Distribution<F> for Gumbel<F>
where
F: Float,
OpenClosed01: Distribution<F>,
{
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> F {
let x: F = rng.sample(OpenClosed01);
self.location - self.scale * (-x.ln()).ln()
}
}

#[cfg(test)]
mod tests {
use super::*;

#[test]
#[should_panic]
fn test_zero_scale() {
Gumbel::new(0.0, 0.0).unwrap();
}

#[test]
#[should_panic]
fn test_infinite_scale() {
Gumbel::new(0.0, f64::INFINITY).unwrap();
}

#[test]
#[should_panic]
fn test_nan_scale() {
Gumbel::new(0.0, f64::NAN).unwrap();
}

#[test]
#[should_panic]
fn test_infinite_location() {
Gumbel::new(f64::INFINITY, 1.0).unwrap();
}

#[test]
#[should_panic]
fn test_nan_location() {
Gumbel::new(f64::NAN, 1.0).unwrap();
}

#[test]
fn test_sample_against_cdf() {
fn neg_log_log(x: f64) -> f64 {
-(-x.ln()).ln()
}
let location = 0.0;
let scale = 1.0;
let iterations = 100_000;
let increment = 1.0 / iterations as f64;
let probabilities = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9];
let mut quantiles = [0.0; 9];
for (i, p) in probabilities.iter().enumerate() {
quantiles[i] = neg_log_log(*p);
}
let mut proportions = [0.0; 9];
let d = Gumbel::new(location, scale).unwrap();
let mut rng = crate::test::rng(1);
for _ in 0..iterations {
let replicate = d.sample(&mut rng);
for (i, q) in quantiles.iter().enumerate() {
if replicate < *q {
proportions[i] += increment;
}
}
}
assert!(proportions
.iter()
.zip(&probabilities)
.all(|(p_hat, p)| (p_hat - p).abs() < 0.003))
}

#[test]
fn gumbel_distributions_can_be_compared() {
assert_eq!(Gumbel::new(1.0, 2.0), Gumbel::new(1.0, 2.0));
}
}
``````