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03_Estimation_part2

# 03_Estimation_part2 - Ch8 p.9 Questions 1 Is the...

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NTHU MATH 2820, 2008, Lecture Notes Ch8, p.9 Questions: 1. Is the estimate (i.e., 24.9) the real λ ? 2. Next time when the same procedure (same sample size, same estimator, same . . . ) is repeated again to get a new estimate, how far the future estimate will be away from 24.9? 3. In which range will you expect say 95% of the future estimate falls? (e.g., [24.8, 25] or [15, 35]?) 4. This is a question related to the stability/uncertainty/ variation of the estimator. 5. How to characterize the stability of an estimator? ( Note. We have to answer the question .) To evaluate the stability/uncertainty of an estimation procedure, it is required to know what the sampling distribution is. made by Shao-Wei Cheng (NTHU, Taiwan) Ch8, p.10 Example 6.5 (cont. Ex.6.4, TBp. 262) Exact sampling distribution of ˆ λ : Because X 1 , X 2 , . . . , X n i.i.d. P ( λ ), S = n i =1 X i P ( n λ ) E( ˆ λ ) = 1 n E ( S ) = λ , Var( ˆ λ ) = 1 n 2 Var( S ) = λ n . Thus, (1) the sampling distribution of ˆ λ is 1 n P ( n λ ), (2) ˆ λ is unbiased, and (3) the standard error of ˆ λ is σ ˆ λ = λ /n. Estimated standard error of ˆ λ is s ˆ λ = ˆ λ /n = 24 . 9 / 23 = 1 . 04 . Asymptotical method : By CLT, sampling distribution of ˆ λ is ap- proximately normal when n is large enough. Since a normally dis- tributed random variable is very unlikely to be more than 2 standard deviation away from its mean, errors in ˆ λ is very unlikely to be more than 2.08.

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NTHU MATH 2820, 2008, Lecture Notes Ch8, p.11 function realization point estimation sampling distribution simulation histogram x 1 , . . . , x n (values) X 1 , . . . , X n (r.v.’s) joint cdf F X 1 ,...,X n ( ·| θ ) ˆ θ 0 (value) ˆ θ (r.v.) θ (parameter) cdf F ˆ θ ( ·| θ ) cdf F ˆ θ ( ·| ˆ θ 0 ) estimated standard error σ ( ˆ θ 0 ) standard error σ ( θ ) Normal cdf exact distribution the form of F ˆ θ ( ·| θ ) is known asymptotical method the form of F ˆ θ ( ·| θ ) is close to Nor- mal cdf (usually when n is large) simulation method used when the form of F ˆ θ ( ·| θ ) is un- known made by Shao-Wei Cheng (NTHU, Taiwan) Ch8, p.12 Example 6.6 (Normal distribution, TBp. 263) The fi rst and second moments for N ( μ , σ 2 ) are μ 1 = E ( X ) = μ μ 2 = E ( X 2 ) = μ 2 + σ 2 μ = μ 1 σ 2 = μ 2 μ 2 1 Let X 1 , X 2 , . . . , X n be i.i.d. N ( μ , σ 2 ) , then the method of moment estimators of μ and σ 2 are ˆ μ = X ˆ σ 2 = 1 n n i =1 X 2 i X 2 = 1 n n i =1 ( X i X ) 2 Sampling distribution of X is Normal( μ , σ 2 n ) and sampling distribution of ˆ σ 2 is σ 2 n χ 2 n 1 . Furthermore, X and ˆ σ 2 are independent.
NTHU MATH 2820, 2008, Lecture Notes Ch8, p.13 Example 6.7 (Gamma distribution, TBp. 263-264) The fi rst two moments of the Γ ( α , λ ) are μ 1 = α / λ μ 2 = α ( α + 1) / λ 2 λ = μ 1 / ( μ 2 μ 2 1 ) α = λμ 1 = μ 2 1 / ( μ 2 μ 2 1 ) Let X 1 , X 2 , . . . , X n be i.i.d. Γ ( α , λ ) , then the method of moment estimators of λ and α are ˆ λ = X/ ˆ σ 2 , ˆ α = X 2 / ˆ σ 2 where ˆ σ 2 = ˆ μ 2 ˆ μ 2 1 .

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03_Estimation_part2 - Ch8 p.9 Questions 1 Is the...

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