# 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 $y$:

 $h(y)=\lim_{\Delta y\rightarrow 0}\frac{P(y\leq Y\leq y+\Delta y\mid Y>y)}{% \Delta y}.$

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

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

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

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 $h(y)=\frac{1}{\beta}\operatorname{exp}(\frac{y-\alpha}{\beta})$.

Title hazard function HazardFunction 2013-03-22 14:27:45 2013-03-22 14:27:45 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 62N99 msc 62P05 cumulative hazard function