I have derived a likelihood function for $\theta$ as follows:
Where $\theta$ is an unknown parameter, $n$ is the sample size, and $s$ is a summary of the data. I now am trying to show that $s$ is a sufficient statistic for $\theta$.
In Wikipedia the Fischer-Neyman factorization is described as:
My first question is notation. In my problem I believe what wikipedia represents as $x$, is $\theta$, and what wikipedia represents as $\theta$ is $s$. Please confirm that that sounds right, it's a point of confusion for me.
Which would mean I'm trying to define the following 3 functions to complete the factorization and confirm that $s$ is sufficient for $\theta$
$$T(\theta)$$ $$g_s(T(\theta))$$ $$h(\theta)$$
But by this point I feel like I've done something wrong, and I'm not really understanding why this factorization is demonstrating sufficiency. I don't really see what is going on with $g$ and $T$.