In the goodness of fit test of Poisson distribution, the degrees of freedom should be $$k - p - 1$$ which means,
ν = (number of categories after pooling) − (number of parameters estimated) − 1
. for example, if I have datatable of 5 categories, and I'm testing if this data is on poisson distribution with the mean of $\lambda$. Then the degrees of freedom would be 3.
But why?? I can't find the proof or explanation of why it should be like this, from nowhere. I tried to find it for all day but couldn't find. So I request you for help. Why is it? or where is the proof?
Thank you!
Suppose we have 5 categories. We derive two things from it. first) poisson mean $\lambda$ from it. second) we derive 'expected frequency' from it, summation of it makes 100%. So, it means we have two simultaneous equations. As we get 5-2(=3) of the Pearson residuals of each categories, we can automatically know the other 2 pearson residuals. Because there're two unknown left, and two simultaneous equations. If you calculate the equations, you get remaining two pearson res. That's why degrees of freedom is 'k-2', not 'k-1'.