[R] Exact poisson confidence intervals
Charles Annis, P.E.
Charles.Annis at StatisticalEngineering.com
Fri Jan 14 01:30:03 CET 2005
Federico:
You might also look at Professor Agresti's observations on exact vs
approximate, which appeared in the American Statistician a few years ago.
(I believe it was the AS; my memory isn't what it once was.)
Google produced this
http://www.amstat.org/publications/tas/index.cfm?fuseaction=agresti1998
when searching for "approximate is better than exact"
Charles Annis, P.E.
Charles.Annis at StatisticalEngineering.com
phone: 561-352-9699
eFax: 614-455-3265
http://www.StatisticalEngineering.com
-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Spencer Graves
Sent: Thursday, January 13, 2005 7:08 PM
To: Uwe Ligges
Cc: R-help mailing list
Subject: Re: [R] Exact poisson confidence intervals
"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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