I'm teaching myself some statistics for fun and I have some confusion regarding sufficient statistics. I'll write out my confusions in list format:
If a distribution has $n$ parameters then will it have $n$ sufficient statistics?
Is there any sort of direct correspondence between the sufficient statistics and the parameters? Or do the sufficient statistics just serve as a pool of "information" so that we can recreate the setting so we can calculate the same estimates for the parameters of the underlying distribution.
Do all distributions have sufficient statistics? ie. can the factorization theorem ever fail?
Using our sample of data, we assume a distribution that the data is most likely to be from and then can calculate estimates (e.g. the MLE) for the parameters for the distribution. Sufficient statistics are a way to be able to calculate the same estimates for the parameters without having to rely on the data itself, right?
Will all sets of sufficient statistics have a minimal sufficient statistic?
This is the material which I am using to try to understand the topic matter here.
From what I understand we have a factorization theorem which separates the joint distribution into two functions, but I do not understand how we are able to extract the sufficient statistic after factorizing the distribution into our functions.
The Poisson question given in this example had a clear factorization, but then it was stated that the sufficient statistics were the sample mean and the sample sum. How did we know that those were the sufficient statistics just by looking at the form of the first equation?
How is it possible to conduct the same MLE estimates using sufficient statistics if the second equation of the factorization result will sometimes depend on the data values $X_i$ themselves? For instance in the Poisson case the second function depended on the inverse of the product of the factorials of the data, and we would no longer have the data!
Why would the sample size $n$ not be a sufficient statistic, in relation to the Poisson example on the webpage? We would require $n$ to reconstruct certain parts of the first function so why is it not a sufficient statistic as well?