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class07-2-handouts

# class07-2-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 07-2- Efficiency, consistency, sufficiency review Jarad Niemi University of California, Santa Barbara 12 May 2010 Jarad Niemi (UCSB) Efficiency, consistency, sufficiency review 12 May 2010 1 / 16

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Class overview Announcements Announcements Homework Homework 5 is returned No need for , δ proofs Show your work State theorems that you use, e.g. Factorization Theorem Mid-term II Friday!! Covers 8.5–8.8,9.1–9.5 Show your work State theorems that you use, e.g. Factorization Theorem Jarad Niemi (UCSB) Efficiency, consistency, sufficiency review 12 May 2010 2 / 16
Class overview Goals Review of estimator properties Efficiency, consistency, sufficiency Example questions Jarad Niemi (UCSB) Efficiency, consistency, sufficiency review 12 May 2010 3 / 16

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Efficiency review Definition Given two unbiased estimators ˆ θ 1 and ˆ θ 2 of a parameter θ , with variances V ( ˆ θ 1 ) and V ( ˆ θ 2 ), respectively, then the efficiency of ˆ θ 1 relative to ˆ θ 2 , denoted eff ( ˆ θ 1 , ˆ θ 2 ), is defined to be the ratio eff ( ˆ θ 1 , ˆ θ 2 ) = V ( ˆ θ 2 ) V ( ˆ θ 1 ) . Definition If ˆ θ is an unbiased estimator of θ , then under very general conditions V ( ˆ θ ) I ( θ ) = nE - 2 log f ( Y ) ∂θ 2 - 1 This is known as the Cram´ er-Rao lower bound . If an estimator has V ( ˆ θ ) = I ( θ ), the estimator is said to be efficient . - E 2 log f ( Y ) ∂θ 2 is often referred to as the Fisher information . Jarad Niemi (UCSB) Efficiency, consistency, sufficiency review 12 May 2010 4 / 16
Efficiency review Definition If ˆ θ is an unbiased estimator of θ , then the efficiency of ˆ θ , denoted eff ( ˆ θ ) is eff ( ˆ θ ) = I ( θ ) V ( ˆ θ ) . Definition If ˆ θ 1 and ˆ θ 2 are biased estimators for θ , then the efficiency of ˆ θ 1 relative to ˆ θ 2 , denoted eff ( ˆ θ 1 , ˆ θ 2 ), is defined to be the ratio eff ( ˆ θ 1 , ˆ θ 2 ) = MSE ( ˆ θ 2 ) MSE ( ˆ θ 1 ) .

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class07-2-handouts - PSTAT 120B Probability Statistics...

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