Hazard Function

Definition

Hazard Function

Let $T\ge 0$ be a non-negative Random Variable representing survival time with PDF $f(t)$ and CDF $F(t)$. The hazard function $0<\lambda (t)<\infty$ of the random variable $T$ is defined as $\lambda (t)=\frac{f(t)}{S(t)}={\lim }_{\Delta t\to 0}\frac{P(T<t+\Delta t|T>t)}{\Delta t}$ where $S(t)$ is the Survival Function.

The hazard function refers to the rate of occurring event at a given time $t$.

Cumulative Hazard Function

The hazard function can alternatively be represented in terms of the cumulative hazard function, defined as $\Lambda (t)={\int }_0^t\lambda (u)du$

Facts

The Survival Function $S(t)$, the Cumulative Hazard Function, the density (PDF) $f(t)$, the Hazard Function, and the distribution function (CDF) of survival time $F(t)$ are related through $S(t)=\exp [-\Lambda (t)]=\frac{f(t)}{\lambda (t)}=1-F(t)$