Re: [R] Fitdistr and MLE for parameter lambda of Poisson distribution

From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>
Date: Sat 11 Feb 2006 - 00:39:19 EST

Gregor Gorjanc <gregor.gorjanc@bfro.uni-lj.si> writes:

> Hello!
>
> I would like to get MLE for parameter lambda of Poisson distribution. I
> can use fitdistr() for this. After looking a bit into the code of this
> function I can see that value for lambda and its standard error is
> estimated via
>
> estimate <- mean(x)
> sds <- sqrt(estimate/n)
>
> Is this MLE? With my poor math/stat knowledge I thought that MLE for
> Poisson parameter is (in mixture of LaTeX code)
>
> l(\lambda|x) \propto \sum^n_{i=1}(-\lambda + x_iln(\lambda)).
>
> Is this really equal to (\sum^n_{i=1} x_i) / n

Yes....

Maximizing l(lambda) is the same as maximizing

sum(x)/n ln lambda - lambda

Now either take the derivative and set equal to zero, or  

rewrite further as equivalent to

ln (lambda/(sum(x)/n)) - (lambda/(sum(x)/n))

and notice that ln(x) - x has a global maximum at x=1 (since ln is strictly concave and the tangent at x=1 is the line y = x - 1)  

(I think this is in the first 20 pages I ever read on theoretical statistics ...)

> --
> Lep pozdrav / With regards,
> Gregor Gorjanc
>
> ----------------------------------------------------------------------
> University of Ljubljana PhD student
> Biotechnical Faculty
> Zootechnical Department URI: http://www.bfro.uni-lj.si/MR/ggorjan
> Groblje 3 mail: gregor.gorjanc <at> bfro.uni-lj.si
>
> SI-1230 Domzale tel: +386 (0)1 72 17 861
> Slovenia, Europe fax: +386 (0)1 72 17 888
>
> ----------------------------------------------------------------------
> "One must learn by doing the thing; for though you think you know it,
> you have no certainty until you try." Sophocles ~ 450 B.C.
>
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-- 
   O__  ---- Peter Dalgaard             ุster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
 (*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard@biostat.ku.dk)                  FAX: (+45) 35327907

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Received on Sat Feb 11 00:44:01 2006

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