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``````// Copyright 2021 Developers of the Rand project.
//
// option. This file may not be copied, modified, or distributed
// except according to those terms.

//! The Fréchet distribution `Fréchet(μ, σ, α)`.

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

/// The [Fréchet distribution](https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution) `Fréchet(α, μ, σ)`.
///
/// The Fréchet distribution is a continuous probability distribution
/// with location parameter `μ` (`mu`), scale parameter `σ` (`sigma`),
/// and shape parameter `α` (`alpha`). It describes the distribution
/// of the maximum (or minimum) of a number of random variables.
/// It is also known as the Type II extreme value distribution.
///
/// # Density function
///
/// `f(x) = [(x - μ) / σ]^(-1 - α) exp[-(x - μ) / σ]^(-α) α / σ`
///
/// # Plot
///
/// The plot shows the Fréchet distribution with various values of `μ`, `σ`, and `α`.
/// Note how the location parameter `μ` shifts the distribution along the x-axis,
/// the scale parameter `σ` stretches or compresses the distribution along the x-axis,
/// and the shape parameter `α` changes the tail behavior.
///
/// ![Fréchet distribution](https://raw.githubusercontent.com/rust-random/charts/main/charts/frechet.svg)
///
/// # Example
///
/// ```
/// use rand::prelude::*;
/// use rand_distr::Frechet;
///
/// let val: f64 = thread_rng().sample(Frechet::new(0.0, 1.0, 1.0).unwrap());
/// println!("{}", val);
/// ```
#[derive(Clone, Copy, Debug, PartialEq)]
#[cfg_attr(feature = "serde", derive(serde::Serialize, serde::Deserialize))]
pub struct Frechet<F>
where
F: Float,
OpenClosed01: Distribution<F>,
{
location: F,
scale: F,
shape: F,
}

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

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

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

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

impl<F> Distribution<F> for Frechet<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()).powf(-self.shape.recip())
}
}

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

#[test]
#[should_panic]
fn test_zero_scale() {
Frechet::new(0.0, 0.0, 1.0).unwrap();
}

#[test]
#[should_panic]
fn test_infinite_scale() {
Frechet::new(0.0, f64::INFINITY, 1.0).unwrap();
}

#[test]
#[should_panic]
fn test_nan_scale() {
Frechet::new(0.0, f64::NAN, 1.0).unwrap();
}

#[test]
#[should_panic]
fn test_zero_shape() {
Frechet::new(0.0, 1.0, 0.0).unwrap();
}

#[test]
#[should_panic]
fn test_infinite_shape() {
Frechet::new(0.0, 1.0, f64::INFINITY).unwrap();
}

#[test]
#[should_panic]
fn test_nan_shape() {
Frechet::new(0.0, 1.0, f64::NAN).unwrap();
}

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

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

#[test]
fn test_sample_against_cdf() {
fn quantile_function(x: f64) -> f64 {
(-x.ln()).recip()
}
let location = 0.0;
let scale = 1.0;
let shape = 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] = quantile_function(*p);
}
let mut proportions = [0.0; 9];
let d = Frechet::new(location, scale, shape).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 frechet_distributions_can_be_compared() {
assert_eq!(Frechet::new(1.0, 2.0, 3.0), Frechet::new(1.0, 2.0, 3.0));
}
}
``````