Struct Loglogistic 

pub struct Loglogistic { /* private fields */ }

Loglogistic distribution with shape α and scale θ: F(x) = (x/θ)^α / (1 + (x/θ)^α), as SciPy’s fisk(c=α, scale=θ) and actuar’s dllogis(shape, scale). Its median is θ.

It is also the growth curve of Clark’s LDF method, G(t) = t^ω / (t^ω + θ^ω), so a fitted curve is this distribution’s cdf.

The tail is Pareto-like: E[X^j] is finite only for j < α, so the mean is infinite for α <= 1 and the variance for α <= 2. Limited and layer moments are finite for every α. With v = F(u), E[min(X, u)^j] = θ^j B(1 + j/α, 1 - j/α; v) + u^j (1 - v), using the unnormalized incomplete beta, which for j >= α has a non-positive second argument and is reached by recurrence.

use prospicio_prob::{Distribution, Loglogistic, Severity};

// Shape 1: F(x) = x / (x + θ) and E[min(X, u)] = θ ln(1 + u/θ).
let d = Loglogistic::new(1.0, 2.0).unwrap();
assert!((d.cdf(3.0) - 0.6).abs() < 1e-15);
assert!((d.lev(3.0) - 2.0 * 2.5f64.ln()).abs() < 1e-13);
assert!(d.mean().is_infinite());

Implementations

impl Loglogistic

pub fn new(shape: f64, scale: f64) -> Result<Self>

Loglogistic with shape α > 0 and scale θ > 0.

pub fn shape(&self) -> f64

Shape α.

pub fn scale(&self) -> f64

Scale θ, the median.

Trait Implementations

impl Clone for Loglogistic

fn clone(&self) -> Loglogistic

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Copy for Loglogistic

impl Debug for Loglogistic

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Distribution for Loglogistic

fn quantile(&self, p: f64) -> Result<f64>

θ (p / (1 - p))^(1/α).

fn mean(&self) -> f64

Expected value.

fn variance(&self) -> f64

Variance.

fn cdf(&self, x: f64) -> f64

P(X <= x).

fn survival(&self, x: f64) -> f64

P(X > x). Representations with a direct form override the default 1 - cdf(x), which loses all precision far in the tail.

fn std_dev(&self) -> f64

Standard deviation.

fn sample(&self, rng: &mut StreamRng, n: usize) -> Vec<f64>

n draws from stream rng, by inverse transform unless the family overrides it (crate::Gamma draws by Marsaglia and Tsang).

fn is_parallel_safe(&self) -> bool

Whether the distribution may be evaluated on several threads at once. True for every native family; false for a crate::Custom whose callbacks must stay on the calling thread (an R function), so the parallel simulations run single-threaded when they meet one.

impl From<Loglogistic> for Dist

fn from(d: Loglogistic) -> Self

Converts to this type from the input type.

impl PartialEq for Loglogistic

fn eq(&self, other: &Loglogistic) -> bool

Equality operator ==.

fn ne(&self, other: &Rhs) -> bool

Inequality operator !=.

impl Severity for Loglogistic

fn stop_loss(&self, retention: f64) -> f64

From the tail, so it does not cancel against the mean; infinite for α <= 1.

fn layer(&self, limit: f64, attachment: f64) -> f64

∫_a^(a+l) S(x) dx, by the tail series in the tail.

fn layer_second_moment(&self, limit: f64, attachment: f64) -> f64

2 ∫_a^b (x - a) S(x) dx with b = a + limit.

fn lev(&self, limit: f64) -> f64

Limited expected value E[min(X, limit)].

fn layer_variance(&self, limit: f64, attachment: f64) -> f64

Variance of the loss to the layer limit xs attachment.

impl StructuralPartialEq for Loglogistic