Discussion
Which of the following theorems is useful for obtaining a sufficient statistics ?
Answer: B
The theorem used to obtain sufficient statistics is the Neyman factorization theorem.
It states that T(X) is sufficient for \(\theta\) if and only if the joint density factorises as \(f(x; \theta) = g(T(x); \theta)\, h(x)\), with \(h\) free of \(\theta\). Reading off such a factorisation is the standard way to find a sufficient statistic.
The Neyman-Pearson lemma gives most powerful tests. Basu's theorem uses a complete sufficient statistic to prove independence from ancillary statistics, and the Rao-Blackwell theorem improves an estimator by conditioning on a sufficient statistic; both take sufficiency as given rather than finding it.
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