Timeline for Is every discrete compound Poisson distribution a mixed Poisson distribution?
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| Dec 5, 2022 at 5:05 | comment | added | Michael Hardy | So the next question would be whether the intersection of these two distinct sets of probability distributions contains anything other than Poisson distributions. | |
| Nov 30, 2022 at 15:24 | comment | added | tparker | @MichaelHardy I mean a discrete compound Poisson process, for which (as I said elsewhere in the question) the random variable $X_n$ is assumed to be natural-number-valued. As I said in the comments, I don't think it matters whether or not we include 0 in the set of natural numbers. I've edited the question to clarify that I'm only considering discrete compound Poisson distributions. | |
| Nov 29, 2022 at 4:45 | comment | added | Michael Hardy | @tparker : BEGIN QUOTE Any compound Poisson process can be canonically rewritten as another equivalent compound Poisson process (potentially with a different rate parameter) in which the compounded R.V. is only supported on the strictly positive integers. END QUOTE I have taken the phrase "compound Poisson distribution" to mean the distribution of $\sum_{n=1}^N X_n$ where $N$ is Poisson-distributed and $(X_i)_{i\,=\,1}^\infty$ are i.i.d. real-valued random variables, not necessarily integer-valued. Your argument seems to work only when they are integer-valued. $\qquad$ | |
| Jul 23, 2022 at 19:52 | comment | added | tparker | So you're certainly correct that if we allow the compounded R.V. to take on negative integer values, that the resulting compound Poisson process isn't a mixed Poisson process. But I'm wondering about if we restrict the compounded R.V. to only take on nonnegative (or positive) values. | |
| Jul 23, 2022 at 19:21 | comment | added | tparker | Hm. I see your argument, but I was under the impression that any compound Poisson process (other than the simplest one where $X \equiv 1$) is over-dispersed. Any compound Poisson process can be canonically rewritten as another equivalent compound Poisson process (potentially with a different rate parameter) in which the compounded R.V. is only supported on the strictly positive integers. In this case $\tau^2 \geq \mu$, so your possibility can't occur. I'll need to think about how to reconcile this argument with yours (which contradicts mine, but I can't see any error on your part). | |
| Jul 23, 2022 at 18:57 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jul 23, 2022 at 18:09 | comment | added | Carlo Beenakker | thanks; I mixed up second moment with variance; hopefully now it's correct. | |
| Jul 23, 2022 at 18:09 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jul 23, 2022 at 17:49 | comment | added | tparker | But isn't a standard Poisson process just a compound Poisson process where the compounded variables are constant and equal to 1, so they have variance $\sigma^2 = 0$? Your argument would then imply that a standard Poisson process has zero variance at any time, which doesn't seem right. | |
| Jul 23, 2022 at 17:31 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jul 23, 2022 at 17:21 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |