hazard function

Let $Y$ be a random variable with probability density function $f_Y(y)$. Then the hazard function $h(y)$ is defined to be:

$$h(y)=\frac{f_Y(y)}{1-F_Y(y)}=\frac{f_Y(y)}{S(y)},$$

where $S(y)$ is the survivor function and $Y$ is the survival time.

The hazard function is the rate of probability of death (non survival) is changing at time $Y=y$, given survival up to time $y$:

$$h(y)=\underset{\mathrm{\Delta }y\to 0}{lim}\frac{P(y\le Y\le y+\mathrm{\Delta }y\mid Y>y)}{\mathrm{\Delta }y}.$$

The cumulative hazard function, $H(y)$ of $Y$ is defined as

$$H(y)={\int }_{-\mathrm{\infty }}^{y}h(t)𝑑t.$$

From this definition, we see that $H(y)=-\ln S(y)$.

Examples.The hazard functions for the three most widely used probability density functions for survival time are:

• The exponential distribution, with $h(y)=\gamma$.

• The Weibull distribution, with $h(y)=\gamma y^{\gamma -1}$ using the standard Weibull distribution.

• The extreme-value distribution, with $h(y)=\frac{1}{\beta }\exp (\frac{y-\alpha }{\beta })$.