Thread: factorization criterion and sufficient statistics

Hi,
I am trying to find a sufficient statistic for p, in X~NB(r,p), where r is a constant.

I know I have to use the factorization criterion, meaning finding the joint pdf and then seperating the joint pdf into g(S,p) and h(x,....,x) (All x values..)

I am having trouble with this, any help appreciated!

So...write $\displaystyle \prod_{i = 1} ^ n f(x_i | p)$ and see what you get. If put that expression in its most logical form, for the negative binomial, it will be obvious what to take as the sufficient statistic. It's the same a common statistic you get with these exponential family distributions (e.g. normal, poisson, binomial, exponential, etc).

So sorry, I haven't learnt how to use the font yet...

For , [COLOR=rgb(0, 0, 0)]I[/COLOR] got (x-1 choose r-1)^n times p^SUMr times (1-p)^ [SUM(x)-SUM(r)].

Anyway, for my g(S,p) i.e. the parameters that have to do with S and p, I got

g(S,p) = p^SUM(r) times (1-p)^(S-SUM(r))

I understand if you can't understand this. I will learn to use the proper language and get back to you hopefully.

Thanks for replying