Is there a goodness of fit test for testing a series of counts against the whole family of poisson distributions? So, with the mean unspecified? Or does that not make any sense?
Kinda related to this, if I know the variable has a poisson distribution, will the poisson distribution with
mean = sample mean always have the best fit (by something like Chi-squared for example) or are there more reliable methods to estimate the mean when your data is highly granular?
Edit: I really don't know what I'm doing, so I'll try to clarify some more just in case there might be some inkling of sense in this question.
Say I counted the number of letters received per day over 30 days and the rate is really low, like 1 letter every 6 days.
The first part of my question is basically: Is it possible to measure how confident I should be in assuming that the variable has a Poisson distribution?
And the second part of my question:
Now, assuming this does follow a Poisson distribution, I could estimate the mean number of letters per day to be $\mu=1/6$ and then do a $\chi^2$ goodness-of-fit test to see how well the data fits the Poisson distribution with this mean.
However, would this be the Poisson distribution with the best fit? Or could there be a Poisson distribution with a different mean which results in a better fit? If so, how would I go about finding such a distribution?