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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?

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closed as too broad by kjetil b halvorsen, Peter Flom Dec 19 '18 at 11:10

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ This is rather a jumble of not-too-closely related questions that seem to be concerned about estimates, counts, and fitting distributions. As such they will be difficult to decipher or answer. What is really going on? What kind of "highly granular" data do you have and what are you trying to learn from them? $\endgroup$ – whuber Feb 14 '15 at 21:50
  • $\begingroup$ @whuber I have absolutely no statistics experience, just working off of the embarrassingly limited exposure I had during undergrad. My question doesn't apply to any specific dataset, but I've always worried about the "how sure am I this variable has poisson distribution"? For example, it could be that some of my letters got lost in the mail, and this depended on the current rate of letters per day, thus (I think) making the distribution non-binomial? $\endgroup$ – new-to-stats Feb 14 '15 at 22:35
  • $\begingroup$ Thank you for making the edit--it helps people understand what you need to know. $\endgroup$ – whuber Feb 14 '15 at 22:35
  • $\begingroup$ I'm voting to close this question as off-topic because it is old, hard to answer (unclear) and the OP seems to have abandoned this site. $\endgroup$ – kjetil b halvorsen Dec 18 '18 at 22:24