Concept

Weibull Distribution

The Weibull distribution generalizes the exponential by adding a shape parameter that controls hazard-rate behavior. Decreasing hazard (early-claim concentration), constant hazard (exponential), and increasing hazard (aging-style risks) all fit naturally inside a single Weibull family.

Page Contract

Role: Concept
Level: Core
Time: Reference
Freshness: Stable

Search Intent

Weibull distribution

Definition And Parameterization

The SOA Loss Models Weibull has shape τ > 0 and scale θ > 0. The support is x > 0. The PDF and CDF have compact closed forms; the survival function is especially clean and is the easiest entry point.

Some statistics texts write the scale as λ in rate form. The SOA convention is the scale form; mixing the two changes the moment expressions.

PDF

f(x)=\frac{\tau}{\theta}\left(\frac{x}{\theta}\right)^{\tau-1}\exp\!\left(-(x/\theta)^{\tau}\right)

Survival function

S(x)=\exp\!\left(-(x/\theta)^{\tau}\right)

Mean and variance

E[X]=\theta\,\Gamma\!\left(1+\tfrac{1}{\tau}\right),\quad \operatorname{Var}(X)=\theta^{2}\Bigl[\Gamma\!\left(1+\tfrac{2}{\tau}\right)-\Gamma^{2}\!\left(1+\tfrac{1}{\tau}\right)\Bigr]

The Shape Parameter Controls Hazard

The hazard rate is h(x) = f(x)/S(x) = (τ/θ)(x/θ)^{τ-1}. When τ = 1, h is constant and the Weibull reduces to an exponential. When τ < 1, h is decreasing in x — the longer a claim has been outstanding, the lower the instantaneous chance of resolution. When τ > 1, h is increasing — aging or wear-out behavior.

Picking Weibull over gamma is usually a hazard-rate choice. Gamma has a hazard that approaches a constant as x grows; Weibull has a hazard that can grow or shrink without bound. If the empirical data suggests a monotonic hazard, Weibull is preferable.

Hazard rate

h(x)=\frac{\tau}{\theta}\left(\frac{x}{\theta}\right)^{\tau-1}

Maximum Likelihood Estimation

With shape τ known, the Weibull scale MLE has a clean closed form. Given observations x_1, ..., x_n, the MLE for θ is the τ-th root of the average of x_i^τ.

With both τ and θ unknown, the likelihood equations couple the two parameters and have no closed-form solution. Standard numerical procedure: profile out θ given τ using the formula below, substitute, and maximize the resulting concentrated likelihood in τ.

Scale MLE with known shape

\hat\theta=\left(\frac{1}{n}\sum_{i=1}^{n}x_i^{\tau}\right)^{1/\tau}

Worked Example: Probability And Mean With τ = 0.7

Time-to-resolution of claims is Weibull with τ = 0.7 and θ = 100 days. The probability a claim resolves within 50 days is 1 − exp(−(50/100)^{0.7}) = 1 − exp(−0.616) ≈ 0.460.

The mean is 100 × Γ(1 + 1/0.7) = 100 × Γ(2.429) ≈ 100 × 1.265 ≈ 126.5 days. With τ < 1, the hazard is decreasing, so claims that survive past 50 days are progressively less likely to resolve in any given day.

Worked Example: Scale MLE With Known Shape

Failure times are Weibull with τ = 2 known and θ unknown. Observed failures: 10, 14, 18, 22, 26. Compute x_i^2: 100, 196, 324, 484, 676. Sum is 1,780; average is 356.

MLE for scale is θ̂ = √356 ≈ 18.87. The fitted distribution has mean 18.87 × Γ(1.5) = 18.87 × 0.8862 ≈ 16.72 and variance 18.87^2 × [Γ(2) − Γ^2(1.5)] = 355.9 × [1 − 0.7854] ≈ 76.4.

Worked Example: Choosing Between Weibull And Gamma

A loss-time dataset has empirical mean 50 and empirical standard deviation 70. The CV is 1.4, so a gamma fit would need shape α = 1/CV^2 ≈ 0.51, which gives a decreasing hazard but a hazard that approaches a constant from above.

A Weibull fit with τ̂ < 1 gives a monotonically decreasing hazard. If diagnostic plots (empirical hazard, Q-Q) show the hazard continuing to fall through the tail, the Weibull fit is preferred. If the empirical hazard flattens, gamma is the better choice. This judgment call is exactly what /concepts/kolmogorov-smirnov-anderson-darling/ and /concepts/model-selection-lrt-aic-bic/ formalize.