Fisher Neyman Factorisation Theorem states that for a statistical model for $X$ with PDF / PMF $f_{\theta}$, then $T(X)$ is a sufficient statistic for $\theta$ if and only if there exists nonnegative functions $g_{\theta}$ and $h(x)$ such that for all $x,\theta$ we have that $f_{\theta}(x)=g_{\theta}(T(x))(h(x))$.
Computationally, this makes sense to me. However, recently I have started to have some doubts about when and where I can apply this theorem.
For example, if I have the PDF for a uniform distribution $f_{\theta}(x)=\frac{1}{\theta}$, doesn't this allow me to make any sufficient statistic that I like by rewriting the PDF as $$f_{\theta}(x)=\Big(\frac{X^2+X^4}{\theta}\Big)\Big(\frac{1}{X^2+X^4}\Big)$$ which would make our sufficient statistic $T(X)=X^2+X^4$. Or if we replace this particular choice of $T(X)$ for something else, doesn't this allow us to construct almost any choice of $T(X)$ as a valid sufficient statistic?
Is this correct, or am I doing something wrong here by arbitrarily adding in functions of $X$ that cancel out in order to create sufficient statistics?
Best Answer
The PDF of the uniform distribution is actually $f_\theta(x) = \frac{1}{\theta} \mathbf{1}_{[0, \theta]}(x)$. This indicator term is important because it ties $x$ and $\theta$ together, and prevents you from doing your cancellation.