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

[Gregor Gorjanc]

Peter Dalgaard wrote:
> 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 ...)

Thank you very much for this. It shows, how much I still need to learn.

-- 
Lep pozdrav / With regards,
    Gregor Gorjanc

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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

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