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When might error terms be Poisson distributed?

I would assume that error terms for Poisson regression will be Poisson distributed. But I could be wrong. I am just guessing. What do you guys make of this?

Stephan Kolassa
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Alyssa DiCarlo
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  • See here: https://stats.stackexchange.com/questions/225405/glm-conditional-unconditional-distribution/ and here: https://stats.stackexchange.com/questions/140568/glm-with-gamma-distribution-of-errors-negative-residuals/ for comments related to whether it makes much sense to talk about "error terms" in Poisson regression and other GLMs in general – Glen_b Aug 04 '20 at 11:19

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In Poisson regression, the observations are (conditionally) Poisson distributed. The errors with respect to the mean are not: if the predicted Poisson parameter is $\hat{\lambda}=0.1$, then we expect future realizations to be $y\sim\text{Pois}(\hat{\lambda})$ distributed - but errors, i.e., $y-\hat{\lambda}$, will take values of $-0.1,0.9,1.9,\dots$, which is certainly not describable by a Poisson distribution. Why do we model noise in linear regression but not logistic regression? is a recent related thread on logistic as compared to Poisson regression.

I can't recall a situation where errors would be Poisson distributed. Unless you would predict $\hat{y}=0$ for all realizations, which are actually Poisson distributed, $y\sim\text{Pois}(\lambda_y)$. Then your errors are of course Poisson. But that doesn't look like a very helpful way of looking at things.

Stephan Kolassa
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  • Thank you for your response. The reason why I was asking this question is because this was a discussion question. I couldn't find anything about error terms being poisson distributed in my book or on the internet. Thank you for clarifying. – Alyssa DiCarlo Aug 05 '20 at 02:15