assignment 5 (ans)

assignment 5 (ans) - 10/11 p. 1 THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: 10/11 p. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 Probability and Statistics: Foundations of Actuarial Science Assignment 5 Solution 1. (a) Yes (b) 1 ˆ μ , 11.1% 2. (a) ( ) ( ) ∑ ∑ ∑ = = = = = n i i n i i n i i i c c X E c L E 1 1 1 μ μ ( ) ( ) ∑ ∑ ∑ = = = = = = n i i n i i n i i i c c X Var c L Var 1 2 2 1 2 2 1 2 σ σ (b) Suppose L is unbiased, then ( ) 1 1 = ⇔ = ∑ = n i i c L E μ By Cauchy-Schwartz inequality, ( )( ) n c c c n i i n i i n i n i i 1 1 1 1 1 2 2 1 1 2 1 2 ≥ ⇒ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∑ ∑ ∑ ∑ = = = = Therefore ( ) ( ) X Var n c L Var n i i = ≥ = ∑ = 2 1 2 2 σ σ . Hence X has the minimum variance among all the linear unbiased estimator for μ , i.e. it is the BLUE. 3. 64% 5. 2 2 1 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Γ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − Γ n n n 6. (a) Based on the normal sample, we have ( ) 2 2 σ = S E , ( ) 1 2 4 2 − = n S Var σ Therefore ( ) ( ) 1 2 4 2 2 − = = n S Var S MSE σ as is unbiased. 2 S (b) ( ) ( ) 2 2 2 2 2 2 2 2 1 1 1 σ σ σ σ σ n n n S n n E SD E SD bias − = − − = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − = ( ) ( ) 2 4 4 2 2 2 1 2 1 2 1 1 n n n n n S n n Var SD Var σ σ − = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 10/11 p. 2 ( ) ( ) ( ) 2 2 2 2 SD Var SD bias SD MSE + = ( ) ( ) 2 4 2 4 2 2 1 2 1 2 1 n n n n n σ σ σ − = − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = (c) Let , 2 2 ˆ cS = σ ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 ˆ σ σ σ σ σ σ − = − = − = − = c c S cE cS E bias ( ) ( ) ( ) 1 2 ˆ 4 2...
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assignment 5 (ans) - 10/11 p. 1 THE UNIVERSITY OF HONG KONG...

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