Assignment 3- solution

Assignment 3- solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2009-10) Assignment 3 (Sketch Solution) 1. (a) bias( T 1 ) = E [ T 1 ]- μ = ( μ + μ ) / 2- μ = 0 ⇒ T 1 is unbiased. MSE( T 1 ) = Var( T 1 ) + bias( T 1 ) 2 = (2 σ 2 ) / 4 + 0 = μ 2 c 2 v / 2. (b) bias( T 2 ) = E [ T 2 ]- μ = (2 μ ) / (2+ c 2 v )- μ =- c 2 v μ/ (2+ c 2 v ). Thus T 2 is biased unless c v = 0. MSE( T 2 ) = Var( T 2 ) + bias( T 2 ) 2 = (2 μ 2 c 2 v ) / (2 + c 2 v ) 2 + c 4 v μ 2 / (2 + c 2 v ) 2 = μ 2 c 2 v / ( c 2 v + 2). (c) T 2 , for MSE( T 2 ) ≤ MSE( T 1 ). 2. (a) For y > 0, cdf of Y is F Y ( y | μ,σ ) = P ( | Z | ≤ y | μ,σ ) = Z y- y ψ ( μ,σ | z ) dz = Z y-∞ ψ ( μ,σ | z ) dz- Z- y-∞ ψ ( μ,σ | z ) dz. The pdf of Y follows by differentiating the above w.r.t. y and noting that ψ ( μ,σ | - y ) = ψ (- μ,σ | y ). (b) (i) ‘ z ( σ ) ∝ ψ (0 ,σ | z ) = ψ (0 ,σ | y ) ‘ * y ( σ ) = ψ (0 ,σ | y ) + ψ (0 ,σ | - y ) ∝ ψ (0 ,σ | y ). (ii) i Z ( σ ) =- E ( ∂ 2 /∂σ 2 )ln ‘ Z ( σ ) = E [- σ- 2 + 3 Z 2 σ- 4 ] = 2 σ- 2 . (iii) i Y ( σ ) =- E ( ∂ 2 /∂σ 2 )ln ‘ * Y ( σ ) = 2 σ- 2 . No information loss. (iv) Yes, according to FC (applied to ‘ z ( σ )). (c) (i) ‘ ** y ( μ ) = ψ ( μ, 1 | y ) + ψ ( μ, 1 | - y ) = ψ ( μ, 1 | y ) + ψ (- μ, 1 | y ). (ii) i Z ( μ ) =- E ( ∂ 2 /∂μ 2 )ln ψ ( μ, 1 | Z ) = 1. (iii) i Y ( μ ) =- E ( ∂ 2 /∂μ 2 )ln ‘ ** Y ( μ ) = E h 1- 4 Y 2 ( e μY + e- μY ) 2 i < 1. There is information loss. 3. (a) i ( λ ) =- E £ ( ∂ 2 /∂λ 2 )ln ‘ ( λ ) / = E ∑ j X j /λ 2 = n ( n + 1)(2 λ )- 1 . (b) ( ∂/∂λ )ln ‘ ( λ ) = 0 ⇒ mle ˆ λ n = 2 ∑ j X j n ( n + 1) . (Second derivative < 0 confirms maximality.) (i) bias = 2 n ( n +1) ∑ j ( jλ )- λ = 0, (ii) variance = 4 n 2 ( n +1) 2 ∑ j ( jλ ) = 2 λ n ( n +1) , (iii) mean squared error = variance. 1 4. Note that the Fisher information i n contained in an i.i.d. random sample of n observations is simply n times the Fisher information i 1 contained in a single observation, i.e. i n = ni 1 ....
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This note was uploaded on 05/04/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.

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Assignment 3- solution - THE UNIVERSITY OF HONG KONG...

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