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class06-1-handouts

# class06-1-handouts - PSTAT 120B Probability Statistics...

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PSTAT 120B - Probability & Statistics Class # 06-1- Efficiency Jarad Niemi University of California, Santa Barbara 3 May 2010 Jarad Niemi (UCSB) Efficiency 3 May 2010 1 / 14

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Class overview Announcements Announcements Homework Homework 4 due today by 4pm in SH 5521 Do not turn homework into my mailbox - you will now get a zero Mid-term II Friday 14 May Personal matter Jarad Niemi (UCSB) Efficiency 3 May 2010 2 / 14
Class overview Last time Intuition -2 -1 0 1 2 2 4 6 8 10 mn 1:n.cis Jarad Niemi (UCSB) Efficiency 3 May 2010 3 / 14

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Class overview Last time Consider Y 1 , . . . , Y n iid N ( μ, σ 2 ). If σ 2 is known then y ± z α/ 2 σ/ n is a 100(1 - α )% confidence interval for μ . If σ 2 is unknown then y ± t n - 1 ,α/ 2 s / n is a 100(1 - α )% confidence interval for μ . Which interval is larger? It turns out E [ S ] = c 4 σ where c 4 = r 2 n - 1 Γ ( n / 2) Γ ([ n - 1] / 2) 1 - 1 4 n - 7 32 n 2 . E t n - 1 ,α/ 2 S / n z α/ 2 σ/ n = t n - 1 ,α/ 2 z α/ 2 σ E [ S ] = t n - 1 ,α/ 2 z α/ 2 σ c 4 σ = t n - 1 ,α/ 2 z α/ 2 c 4 Jarad Niemi (UCSB) Efficiency 3 May 2010 4 / 14
Class overview Goals Efficiency of estimators Cram´ er-Rao lower bound, exercise 9.8 Jarad Niemi (UCSB) Efficiency 3 May 2010 5 / 14

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Efficiency 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
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class06-1-handouts - PSTAT 120B Probability Statistics...

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