Hazard Function

Definition

Hazard Function $h(t)$: Instantaneous failure rate at time $t$ given survival until $t$.

$h(t)=\lim \_\Delta t\to 0\frac{\mathrm{Pr}(t<T\le t+\Delta t\mid T>t)}{\Delta t}=\frac{f(t)}{S(t)}$

About Hazard Function

Rate of the event occurring per unit time(t), among those who haven’t experienced the event yet.

• A high $h(t)$ at some time $t$ means: among those still alive at $t$, failures are happening rapidly.
• A low $h(t)$ means failures are rare at that moment.

The “instantaneous” part means you’re shrinking the window to a single point in time rather than measuring over a finite interval.

Properties

Can be:

• Increasing
• Decreasing
• Constant
• Bathtub-shaped

Interpretation Example

100 light bulbs running at time $t=1000$ hours. 5 fail between 1000 and 1001 hours.

$h(1000)\approx \frac{5/100}{1}=0.05\text{ per hour}$

At $t=1000$, survivors are failing at a rate of 0.05 per hour.

Related

• Survival Function
• Hazard Function Derivation