Weibull Distribution (Survival)

Definition

Weibull Distribution for survival: Generalization of exponential with shape parameter $\alpha$. Can model increasing, decreasing, or constant hazard.

For $T\sim \text{Weibull}(\lambda ,\alpha )$:

$$\begin{array}{rl}f(t)& =\alpha \lambda t^{\alpha -1}{e}^{-\lambda t^{\alpha }},t>0\\ S(t)& =\exp (-\lambda t^{\alpha })\\ h(t)& =\alpha \lambda t^{\alpha -1}\\ H(t)& =\lambda t^{\alpha }\end{array}$$

Property: Shape Parameter Effects

• $\alpha <1$: Decreasing hazard
• $\alpha =1$: Constant hazard (reduces to exponential)
• $\alpha >1$: Increasing hazard

Example

Case: Risk of getting sick during flu season. Hazard increases over time as the season peaks.

Use Weibull with $\lambda =0.001$, $\alpha =2$ (increasing hazard):

$h(t)=\alpha \lambda t^{\alpha -1}=0.002t$

$t$ (days)	$h(t)$
10	0.02
30	0.06
60	0.12

Interpretation: On day 10, survivors are getting sick at 0.02/day. By day 60, that rate jumps to 0.12—the flu season is peaking.

Related

• Exponential Distribution (Survival)
• Parametric Survival Distributions Cheatsheet