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

[Peter Dalgaard]

Gregor Gorjanc <gregor.gorjanc at 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 
> 
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> 
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-- 
   O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
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