Fisher information, sufficiency, and ancillarity: some clarifications

Misconceptions are many when it comes to Fisher information, sufficiency, and ancillarity, especially among beginners. Many believe that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\mathcal I _{T_{1}}(\theta )+\mathcal I _{T_{2}}(\theta )$$</EquationSource> </InlineEquation> should equal <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathcal I _{T_{1}+T_{2}}(\theta )$$</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\theta $$</EquationSource> </InlineEquation>. We exhibit precise scenarios where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathcal I _{T_{1}+T_{2}}(\theta )$$</EquationSource> </InlineEquation> is <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$>, =,$$</EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$>\mathcal I _{T_{1}}(\theta )+\mathcal I _{T_{2}}(\theta )$$</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\theta $$</EquationSource> </InlineEquation>. Then, we clarify the process of verifying ancillarity of a statistic with concrete examples. Copyright Sapienza Università di Roma 2013