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Suppose I have specifically a single observation $x$ which I believe comes from a Poisson distribution. Could I estimate a 95% confidence interval for the true mean as $\sigma * \sqrt{x}$ with $\sigma = 1.96$?

In other words, if I observe a count of $x=10$, can I estimate a 95% confidence interval as $x \pm \sigma * \sqrt{x} = 10 \pm 1.96 \sqrt(10) = (3.801936, 16.198064)$?

IWS
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Nils
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  • Yes, I did see that question. I think this may indeed be equivalent assuming $\lambda = x$ and $n = 1$, but I just wanted to confirm that this is correct. – Nils Aug 02 '17 at 13:03
  • Fair enough. Do note however, that of course you can calculate a mean of one observation, but that there is no true variance in a single measurement... In any case, I'd expect the $x±σ∗x−−√$ confidence interval to be too narrow as it does not take this into account. How to do this I must admit, I do not know. – IWS Aug 02 '17 at 13:08
  • Not sure I am reading your formula correctly, but I would have thought that I do account for $n=1$ because I use $\sqrt{x}$ above instead of $\sqrt{x/n}$? – Nils Aug 02 '17 at 13:13
  • Excuse me, I mixed up the formula lay-out, I tried to copied yours. Yes you do account for $n=1$ somehow, but I do not know whether it is enough (n=1 might be a special case). – IWS Aug 02 '17 at 13:15
  • http://ms.mcmaster.ca/peter/s743/poissonalpha.html suggests using something like the R code c(qchisq(0.025, 2*10)/2, qchisq(0.975, 2*(10+1))/2) to give $(4.795389, 18.390356)$, though this is conservative in the sense of often being more than $95%$ – Henry Aug 03 '17 at 23:49

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