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Assign 3 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II SUGGESTED SOLUTION TO ASSIGNMENT 3 (MARCH 2008) 1. Question 1 (a) Bias( T 1 ) = E ( T 1 )- μ = μ + μ 2- μ = 0 , thus T 1 is unbiased. MSE( T 1 ) = E X 1 + X 2 2- μ ¶ 2 = 1 4 E ( X 1 + X 2- 2 μ ) 2 = 1 4 { E ( X 1- μ ) 2 + 2 E ( X 1- μ ) E ( X 2- μ ) + E ( X 2- μ ) 2 } = σ 2 2 or μ 2 c 2 v 2 or noticing that T 1 is unbiased, just use MSE( T 1 ) = Var( T 1 ) = Var( X 1 + X 2 2 ) = Var( X 1 ) + Var( X 2 ) 4 = μ 2 c 2 v 2 . (b) Bias( T 2 ) = E ( T 1 )- μ = 2 μ 2 + c 2 v- μ =- μc 2 v 2 + c 2 v , thus T 2 is biased unless c v = 0 . MSE( T 2 ) = Var( T 2 ) + { Bias( T 2 ) } 2 = E X 1 + X 2 2 + c 2 v- 2 μ 2 + c 2 v ¶ 2 +- μc 2 v 2 + c 2 v ¶ 2 = 1 (2 + c 2 v ) 2 E ( X 1 + X 2- 2 μ ) 2 + μc 2 v 2 + c 2 v ¶ 2 = 2 σ 2 + μ 2 c 4 v (2 + c 2 v ) 2 = μ 2 c 2 v 2 + c 2 v . (c) T 2 is recommended. A good estimator should be accurate and pre- cise. T 1 is unbiased and T 2 might not, that is, T 1 is more accurate. However, T 2 is preciser since Var( T 1 ) Var( T 2 ) = 1 + 4 c 2 v + c 4 v 4 > 1 or Var( T 1 ) > Var( T 2 ) . Thus we use MSE to measure the qualities of these two estimators since it takes into account both accuracy and precision. And smaller 1 MSE implies the sampling distribution of the corresponding estima- tor is more concentrated near the parameter. In the question, by some algebra, we get MSE( T 1 ) MSE( T 2 ) = 1 + c 2 v 2 > 1 or MSE( T 1 ) > MSE( T 2 ) . . 2. Question 2 (a) For y > 0, by a transformation, cdf of Y is F Y ( y | μ,σ ) = P ( | Z | ≤ y ) = P (- y ≤ Z ≤ y ) = Z y- y ψ ( μ,σ | t ) dt = Z- y ψ ( μ,σ | t ) dt + Z y ψ ( μ,σ | t ) dt = Z y ψ (- μ,σ | t ) dt + Z y ψ ( μ,σ | t ) dt Making the first order differential with respect to y , we prove the conclusion. (b) i. l z ( σ ) = ψ (0 ,σ | z ), l * y ( σ ) = ψ (0 ,σ | y ) + ψ (0 ,σ | - y ) = 2 ψ (0 ,σ | y ), ii. log l z ( σ ) =- log σ- z 2 / 2 σ 2 , d log l z ( σ ) /dσ =- 1 /σ + z 2 /σ 3 , d 2 log l z ( σ ) /dσ 2 = 1 /σ 2- 3 z 2 /σ 4 ,- E [ d 2 log l z ( σ ) /dσ 2 ] = 2 /σ 2 . iii. It is the same as (ii). No information loss. iv. Yes. By factorization criterion, it is sufficient. (c) i. Substituting σ = 1 into (a), we get the conclusion. ii. log ψ ( μ, 1 | y ) =- ( z- μ ) 2 / 2 , d log ψ ( μ, 1 | y ) /dμ = z- μ, i 1 ( μ ) =- E [( d log ψ ( μ, 1 | y ) /dμ ) 2 ] = V ar ( Z ) = 1 iii. log l ** y = log[exp(- ( y- μ ) 2 2 ) + exp(- ( y + μ ) 2 2 )], d log l ** y /dμ = exp(- ( y- μ ) 2 2 )( y- μ )- exp(- ( y + μ ) 2 2 )( y + μ ) exp(- ( y- μ ) 2 2 )+exp(- ( y + μ ) 2 2 ) = exp( μy )( y- μ )- exp(- μy )( y + μ ) exp( μy )+exp(- μy ) , d 2 log l ** y /dμ 2 = [exp( μy )+exp(- μy )] 2- 4 y 2 [exp( μy )+exp(- μy )] 2 Fisher information contained in Y is i 2 ( μ ) = 1- 4 E [ { exp( μY ) + exp(- μY ) }- 2 Y 2 ] < 1....
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Assign 3 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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