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I have empirical data on a process that I assume is Poisson with a given mean, say $\mu$ (unknown). The data is of the form $(x_i, 1\leq i\leq n)$ for $n$ consecutive time periods.

I am concerned that the process remains stable, by which I mean that $\mu$ does not change over time (as these are "negative" events, I am mainly concerned about the mean switching to a higher value for some unforeseen external reason). Therefore I want to be able to detect if a new occurrence ($x_{n+1}$) exceeds what is reasonably expected (for instance if the empirical history is mostly $0$'s and $1$'s, $x_{n+1}=100$ would be unexpected -- although theoretically possible).

I would like to be able to get:

  1. A predictive $1-\alpha$ confidence interval on $x_{n+1}$, either one sided or two-sided, and

  2. A test of the form "if $x_{n+1}\geq$ some value, reject the hypothesis that $x_{n+1}$ is generated by a Poisson process with the same mean as my historical data.

I have seen a number of papers on the estimation of $\mu$ but this is a bit different.

Thanks!

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    $\begingroup$ If you believe your Poisson counts to be independent (though in many situations I might find that surprising), you could try searching for Poisson tolerance intervals $\endgroup$
    – Glen_b
    Aug 3, 2019 at 6:34

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After some research and the help of a comment by @Glen_b I will answer my question. The correct keyword is prediction interval; Wikipedia has this article about the subject.

A recent reference is Improved Closed-Form Prediction Intervals for Binomial and Poisson Distributions (Krishnamoorthy & Peng, 2011) [PDF link].

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