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 : where is a random variable denoting the time of death. Let denotes the pdf of and denote the cdf of . We can see that must be non-increasing since and is nonnegative.
Another fundamental object is the hazard function of , , which gives the rate at which the probability of death is changing at some time : Using the definition of conditional probability and the cdf, we can write
The cumulative hazard function of , , is defined as is also called the integrated hazard. It follows that .
Some commonly used survivor functions (survival time distributions) include:
- The exponential distribution:
- The Weibull distribution:
There is positive or negative duration dependence as .
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