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I am trying to find out what is the distribution governing the inter-arrival of events for a counting process that follows a negative binomial? I know for a Poisson counting process the inter-arrival times follow an exponential distribution, but I can't seem to find the answer for a negative binomial process.

Thank you.

kjetil b halvorsen
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david moutarde
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  • It's going to depend on how the negative binomial itself is generated. For example, one way is to say you have a Gamma distribution on the Poisson parameter; another is to say you have a Poisson distribution for the occurrence of "events", and when an event occurs, you have a logarithmic distribution for the associated count; the distribution of the number of counts is negative binomial, but the interarrival times are a mixture of a discrete mass at zero and an exponential distribution. – jbowman Jan 24 '20 at 15:50
  • See also https://stats.stackexchange.com/questions/77762/negative-binomial-process – kjetil b halvorsen Jan 24 '20 at 15:51
  • Not sure I fully understand the comments. But, the NB is a fit to a dataset I have done on event occurrence. I would like to know simulate and get some simulated event occurrences. What distribution should I use to get those arrival times? – david moutarde Jan 24 '20 at 16:53
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    Knowing that the number of events over some fixed period of time $\sim$ Negative Binomial isn't enough to tell you the interarrival time distribution, because there are different interarrival time distributions that can result in exactly the same Negative Binomial distribution for the # of events over a fixed time interval. – jbowman Jan 24 '20 at 19:20
  • Thanks for the reply. It is getting more clear to me now. So knowing that its a NB, you say there is potentially a range of interarrival time distributions I can use. Could you point me to one that is easy to use and what would be the parameters of that distribution given that we know the NB parameters? Thanks. – david moutarde Jan 27 '20 at 11:19

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