[R] Exact poisson confidence intervals

[Spencer Graves]

      "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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