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Unformatted text preview: February 22 nd Today : Cramer-Rao lower bound Unbiased Estimator of , say 1 2 3 , , , L It could be really difficult for us to compare ( ) i Var when there are many of them. Theorem. Cramer-Rao Lower Bound Let 1 2 , , , n Y Y Y L be a random sample from a population with p.d.f. ( , ) f y . Let 1 2 ( , , , ) n h y y y = L be an unbiased estimator of . Given some regularity conditions (continuous differentiable etc.) and the domain of ( , ) i f y does not depend on . Then, we have 2 2 2 1 1 var( ) (ln ( )) (ln ( )) f f n E n E = - Theorem. Properties of the MLE Let . . . ~ ( , ) , 1,2, , i i d i Y f y i n = L Let be the MLE of Then, 2 1 ( , ) ln( ( )) n N f n E The MLE is asymptotically unbiased and its asymptotic variance : C-R lower bound Example 1. Let . . ....
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This note was uploaded on 11/14/2010 for the course AMS 312 taught by Professor Zhu,w during the Spring '08 term at SUNY Stony Brook.
- Spring '08