# Duration Models

Survival analysis concerns statistical models of the time until the first occurance of some event. In a biological setting, the event in question might be the death of an organism. In mechanics, it could be the failure of a machine. In economics, it could be entering the workforce.

The survivor function is the fundamental part of such models, denoting the probability that the time of death is later than some time $t$: $S\left(t\right)=\mathrm{Pr}\left(T>t\right)$ where $T$ is a random variable denoting the time of death. Let $f\left(t\right)$ denotes the pdf of $T$ and $F\left(t\right)$ denote the cdf of $T$. We can see that $S$ must be non-increasing since $S\left(t\right)=1-F\left(t\right)=1-{\int }_{0}^{t}f\left(s\right)\phantom{\rule{thickmathspace}{0ex}}\mathrm{ds}$ and $f$ is nonnegative.

Another fundamental object is the hazard function of $T$, $\lambda \left(t\right)$, which gives the rate at which the probability of death is changing at some time $t$: $\lambda \left(t\right)\equiv \underset{h\to 0}{\mathrm{lim}}\frac{\mathrm{Pr}\left(t\le T\le t+h|T>t\right)}{h}.$ Using the definition of conditional probability and the cdf, we can write $\lambda \left(t\right)=\frac{f\left(t\right)}{1-F\left(t\right)}=\frac{f\left(t\right)}{S\left(t\right)}.$

The cumulative hazard function of $T$, $\Lambda \left(t\right)$, is defined as $\Lambda \left(t\right)\equiv {\int }_{0}^{t}\lambda \left(t\right)\phantom{\rule{thickmathspace}{0ex}}\mathrm{dt}.$ $\Lambda \left(t\right)$ is also called the integrated hazard. It follows that $\Lambda \left(t\right)=-\mathrm{ln}S\left(t\right)$.

## Common Distributions

Some commonly used survivor functions (survival time distributions) include:

• The exponential distribution:

$F\left(t\right)=1-{e}^{-\gamma t}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\lambda \left(t\right)=\gamma .$

• The Weibull distribution:

$F\left(t\right)=1-{e}^{-\gamma {t}^{\alpha }}\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\lambda \left(t\right)=\gamma \alpha {t}^{\alpha -1}.$

There is positive or negative duration dependence as $\alpha \gtrless 1$.