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:

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The exponential distribution, with $h(y)=\gamma$ .
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The Weibull distribution, with $h(y)=\gamma y^{\gamma-1}$ using the standard Weibull distribution.
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The extreme-value distribution, with $h(y)=\frac{1}{\beta}\operatorname{exp}(\frac{y-\alpha}{\beta})$ .