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Suppose $T_1(X)$ is a sufficient statistic for $\theta$ for a family of distributions indexed by $\theta$ and $T_2(X)$ is another statistic that induces a “finer”(what is meant by a finer ? ) partition than $T_1$. How do I know that $T_2$ is also sufficient ?

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Consider the following partitions: $\textbf{P} = \{[1,2], (2,4] \}$ and $\textbf{P'} = \{[1,2), \{2 \}, (2,3), [3,4] \}$. Then $\textbf{P'}$ is finer than $\textbf{P}$. In other words, for every $J$ in $\textbf{P'}$ there exists a $K$ in $\textbf{P}$ such that $J \subseteq K$. Sufficiency is a property of partitions. So if we have sufficiency over $[1,2]$ we have sufficiency over $[1,2)$ and $\{2 \}$. Likewise, if we have sufficiency over $(2,4]$, we have sufficiency over $(2,3)$ and $[3,4]$. If we look at sufficiency over the coarsest partition $\textbf{Q} = \{[1,4] \}$ then this is the minimal sufficient statistic.

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