Re: [R] Exact poisson confidence intervals

Spencer Graves Thu, 13 Jan 2005 16:11:05 -0800

"Exact confidence limits" are highly conservative. I have not studied this for the Poisson distribution, but for the binomial distribution, Brown, Cai and DasGupta (2001, 2002) showed that the exact coverage probabilities exhibit increasingly wild oscillations as the binomial probability goes to either 0 or 1. The interval width for "exact" 95% confidence intervals is increased to compensate for these oscillations so the minimum coverage is 95%. In practice, this means that the actually coverage may be much higher, possible as much as 99% or more in most applications. Moreover, unless the binomial / Poisson parameter is exactly constant, any minor variations in the parameter would move the peaks to fill the valleys, making the "exact" intervals highly conservative.

As part of this work, Brown, Cai and DasGupta also showed that the actual coverage probabilities of the standard approximate confidence limits [p.bar +/-2*sqrt(p.bar*(1-p.bar)/n)] are highly biased. They described several other alternatives. It turns out that the standard asymptotic normal approximation to the logit actually performs fairly close to the best.

By extension, I would expect that the standard asymptotic normal approximation for the log(PoissonRate) might perform better than other confidence intervals for the Poisson, though of course, this should be verified. At the risk of making a fool of myself, I'll continue with this exercise: If I haven't made a mistake, the Fisher information for g = log(PoissonRate) is the PoissonRate, so the approximate standard deviation for g-hat is 1/sqrt(PoissonRate). But the maximum likelihood estimate for the PoissonRate is x.bar = mean of the Poisson observations. This would suggest x.bar*exp(+/-2/sqrt(x.bar)) as an approximate 95% confidence interval for a Poisson. If someone does any checks on this, I would like to hear the results.

hope this helps. spencer graves ########################### Brown, Cai and DasGupta (2001) Statistical Science, 16: 101-133 and (2002) Annals of Statistics, 30: 160-2001 ###########################

Uwe Ligges wrote:

Federico Gherardini wrote:
Hi all, Is there any R function to compute exact confidence limits for a Poisson distribution with a given Lambda?
For sure you are looking for certain quantiles of the poisson distribution? See ?Poisson.
Uwe Ligges
Thanks in advance
Federico
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Re: [R] Exact poisson confidence intervals

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