Neyman's Factorization Theorem

Definition

Let $X_1,X_2,\dots ,X_n$ be a Random Sample with PDF $f(x|\theta )$, where $\theta \in \Omega$, then The statistic $Y_1=u_1(X_1,X_2,\dots ,X_n)$ is a Sufficient Statistic for $\theta$ if and only if $\exists k_1,k_2,{\prod }_{i=1}^{n}f(x_i|\theta )=k_1(y_1|\theta )k_2(x_1,x_2,\dots ,x_n)$ where $k_1,k_2$ are non-negative functions, and $k_2$ does not depend on $\theta$