Predictive distributions in Poisson regression I have the following question about Poisson regression. In "regular" (OLS) linear regression it is fairly easy to prove that the correct predictive distribution will be a $t$ distribution, but what if I want to use a Poisson regression? Is there any literature on predictive distributions in the Poisson model?
 A: In frequentist statistics, a confidence interval for a parameter may be generated via a pivotal quantity - a function of the data and the parameter whose distribution doesn't depend on the parameter. One then creates an interval for the pivotal quantity, and backs out an interval for the parameter.
Prediction intervals may be similarly treated. In that case, the quantity is a function of an unknown (/'future') observation and the data, whose distribution doesn't depend on the future value. One then creates an interval for the pivotal quantity, and backs out an interval for the future value.
In the case of linear regression $\frac{y_f-\hat{y_f}}{\text{se}(y_f-\hat{y_f})}$ has a t-distribution with $n-p-1$ d.f (where there are $p$ predictors not including the constant).
It would be nice, therefore, to produce an interval for a Poisson value in this way. However, in general, there's no pivotal quantity. 
In a few very simple specific instances, it may be possible to get a suitable quantity.
[For example, if you have a simple constant Poisson rate $\lambda$ per observation-interval, with $n$ iid observations $y_1, y_2, ..., y_n$, and one future observation ($y_{f}$), then $\frac{y_f}{y_f+S}$ for $S=\sum_{i=1}^{n} y_i$ will be a binomial proportion; I think you can use that sort of information to back out an interval for $y_f$, and I believe the interval is equivalent to a negative binomial for $y_f|S$.]
However the general case doesn't work. 
In some cases you can do some reasonable approximate intervals.
This paper (Krishnamoorthy & Peng, Improved Closed-Form Prediction Intervals for
Binomial and Poisson Distributions) discusses Poisson prediction intervals (but not from a pivotal approach).
