Lecture5

# Lecture5 - AMS312.01 Spring 2010 Professor Wei Zhu Stony...

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Unformatted text preview: AMS312.01 Spring 2010 Professor Wei Zhu Stony Brook University February 15 th Chapter 5: Point Estimation Point Estimators Example. Let 1 2 , , , n X X X L be a random sample from 2 ( , ) N μ σ . Please find a good point estimator for 1. μ 2 2. σ Solutions. ˆ 1. X μ = 2 2 ˆ 2. S σ = There are the typical estimators for μ and 2 S . Both are unbiased estimators . Property of Point Estimators Unbiased. ˆ θ is said to be an unbiased estimator for θ if ˆ ( ) E θ θ = . 1 2 ( ) n X X X E X E n + + + = L 1 2 ( ) ( ) ( ) n E X E X E X n n μ μ μ μ + + + + + + = = = L L 2 2 ( ) E S σ = (Proof: Homework #2 extra problem) Unbiased estimator may not be unique. Example. ( 29 [ ] i i i E a X a μ = ∑ ∑ 1 1 since ( ) n i i i n i i a X E a μ μ μ = = = = ∑ ∑ % % 1 AMS312.01 Spring 2010 Professor Wei Zhu Stony Brook University Variance of the unbiased estimators 2 2 ( ) ( ) ( ) when 1 ( ) i i Var X Var X Var X n n Var X σ σ = = f p Methods for deriving point estimators 1. Maximum Likelihood Estimator (MLE) 2. Method of Moment Estimator (MOME) Example (continued). . . . 2 ( , ) , 1, , ~ i i d i X N i n μ σ = L 1. Derive the MLE for 2 and μ σ . 2. Derive the MOME for 2 and μ σ . Solution. 1. MLE [i] pdf: 2 2 ( ) 1 2 2 2 2 2 2 ( ) 1 ( ) (2 ) exp , , 1, , 2 2 i x i i i x f x e x R i n μ σ μ πσ σ πσ--- - = =- ∈ = L [ii] likelihood function 1 2 1 ( , , , ) ( ) n n i...
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Lecture5 - AMS312.01 Spring 2010 Professor Wei Zhu Stony...

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