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I'm analyzing a noisy time series where where the inter-event interval is known to follow a two-gamma mixture distribution. If there was a simple model that would generate that kind of thing, it would be pretty simple to implement into BUGS. But otherwise, I can't think of anything that wouldn't be prohibitively kludgey.

Can anyone think of a model that induces mixture-gamma waiting times?

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    $\begingroup$ Psst... you might want to ask a question in your question. $\endgroup$ Commented Nov 22, 2010 at 21:45
  • $\begingroup$ Heh, sorry about that. $\endgroup$
    – DavidShor
    Commented Nov 22, 2010 at 22:03

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A Gamma distribution with an integer shape parameter is an Erlang distribution which in turn is a generalization of the Exponential distribution often used to model waiting times. Thus, perhaps one approach is to use an Erlang distribution instead of a Gamma to model your waiting times. One way to justify the Erlang distribution is to assume that inter-event times are triggered by a certain number of underlying exponential inter-event times.

For example, think of water droplets falling into a balloon at some unknown rate. As water droplets keep falling the balloon will pop at some point. The waiting time for a balloon to pop will be erlang distributed as long as the inter-arrival times of the water droplets are exponentially distributed.

A mixture erlang distribution can have the following interpretation: The inter-event times are now dependent on two separate exponentially distributed processes with perhaps different parameters.

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  • $\begingroup$ Thank you! Is that an asymptotic result? How high does the trigger have to be? $\endgroup$
    – DavidShor
    Commented Nov 23, 2010 at 22:45
  • $\begingroup$ A sum of $k$ exponentials with parameter $\lambda$ gives the erlang with parameters $k$ and $\lambda$. Thus, it is not a asymptotic result but a finitary one. $\endgroup$
    – user28
    Commented Nov 24, 2010 at 0:20

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