Fitdistr and MLE for parameter lambda of Poisson distribution

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 ...)