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So from my understanding, the hazard rate is not a probability but a rate of an event happening at any one period of time. Can someone please tell me the implications of a hazard rate value. Would anyone be able to provide a question that is possible to only answer with the hazard rate?

And would anyone please verify the following: the hazard rate is simply the pdf of some point in time in the future assuming the get to that point in time

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  • $\begingroup$ possible duplicate of What is a hazard rate? $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Feb 2, 2015 at 16:51
  • $\begingroup$ See also here & here. $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Feb 2, 2015 at 16:55
  • $\begingroup$ No not really. I understand that it is not a probability and I have read up on that question. I would actually prefer an example question and my specific questions answered. But thanks for pointing that out andy $\endgroup$
    – alaboudi
    Feb 2, 2015 at 16:56
  • $\begingroup$ So your question's "What's the use of expressing a probability distribution as a hazard function?"? (That hazard's a conditional density is seen from the definition here.) $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Feb 2, 2015 at 17:15
  • $\begingroup$ Scortchi please provide me a question where the answer would be the output of the hazard rate function? $\endgroup$
    – alaboudi
    Feb 2, 2015 at 18:31

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Don't think of hazard as a density function. It is an arrival rate, a probability per unit time.

I always think of hazard in terms of a conditional probability tree. I think of the hazard as the expected number of events in a unit of time, and it has a Poisson distribution. Double the time, the expected number of events doubles. Halve the time, the expected number of events halves. If I make the time step small enough, the expected number of events is either 0 (survival) or 1 (death). If it is 0, the coin gets to flip again.

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Here, if the time is in discrete days, on each day there is a 0.1 chance of death, and a 0.9 chance of surviving until tomorrow. That's a hazard of 0.1 death per day. The probability of surviving to time $t$ is $0.9^t$, obviously, or $(1 - 0.1)^t$, obviously.

Rather than multiplying everything, you can add their logs, and the log of 1-P is just -P, if P is small enough. You can make it small enough by shortening the intervals.

If you cut the time steps in half, you also cut the probability of death in half, so the tree gets smoother but keeps about the same shape. In the limit it is $exp{(-h t)}$. Of course, the total probability of dying at that time is the probability of surviving to that time, multiplied by the local probability of dying at that time, which is now $P(death) * deltaTime$, or $h$, which is now a density, as you can see, or $h*exp(-h t)$.

Notice that hazard $h$ does not have to be constant over time. You can replace $h t$ by its time-integral, and you still get a correct survival function.

Thinking of it as a conditional probability tree makes it really easy (for me) to understand things like right-censoring, interval-censoring, etc.

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