Fisher information, sufficiency, and ancillarity: some clarifications

摘要

Misconceptions are many when it comes to Fisher information, sufficiency, and ancillarity, especially among beginners. Many believe that (mathcal I _{T_{1}}(theta )+mathcal I _{T_{2}}(theta )) should equal (mathcal I _{T_{1}+T_{2}}(theta )) for all (theta ). We exhibit precise scenarios where (mathcal I _{T_{1}+T_{2}}(theta )) is (<, =,) or (>mathcal I _{T_{1}}(theta )+mathcal I _{T_{2}}(theta )) for all (theta ). Then, we clarify the process of verifying ancillarity of a statistic with concrete examples.