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I have reduced the estimation problem of a real event (a technical failure happening, the fact of which is checked in regular time intervals) to the following problem:

we have a non-fair coin, which gives a head for almost every throw. It gives a tail extremely rarely. Let's say we throw the coin every second. After an hour, a tail comes up. After two minutes, another. After 2 hours, another. How can we estimate the probability of a tail occurring, and how can we have a good guess at the reliability of the estimate after a given number of (e.g. few dozen) occurrences of a tail?

My problem is that the event is rare enough that it's very hard to get a reliable measurement for some of the longest time periods when it doesn't occur (it just takes a lot of time).

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  • $\begingroup$ What about a simple exponential distribution for the time between failures? $\endgroup$
    – Matteo Fasiolo
    Commented Nov 28, 2013 at 13:15
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    $\begingroup$ Have you looked into survival analysis? $\endgroup$
    – wvguy8258
    Commented Nov 28, 2013 at 15:27

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Could be a Binomial B(n,p) with n very large and p very small. In such cases you can also have a look to the "law of rare events" which is a Poisson(np) simpler to manage than a Binomial. Look at https://en.wikipedia.org/wiki/Poisson_distribution#Law_of_rare_events

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