HW8%20Solutions - AD Number Name Score IE 300 GE 331 HW8...

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AD Number________ Name_________________________ Score________ IE 300 / GE 331 HW8 7-18 (1 point) Θ 1 , Θ 2 are unbiased, Θ 3 is biased 𝑀𝑀𝑀𝑀𝑀𝑀�Θ 1 = 𝑉𝑉𝑉𝑉𝑉𝑉�Θ 1 = 12, 𝑀𝑀𝑀𝑀𝑀𝑀�Θ 2 = 𝑉𝑉𝑉𝑉𝑉𝑉�Θ 2 = 10, 𝑀𝑀𝑀𝑀𝑀𝑀�Θ 3 = 6 We select Θ 3 since it has the smallest MSE 7-26 (4 points) (a) 𝑀𝑀 ( 𝜇𝜇̂ ) = 𝛼𝛼𝑀𝑀 ( 𝑋𝑋 1 ) + (1 − 𝛼𝛼 ) 𝑀𝑀 ( 𝑋𝑋 2 ) = 𝛼𝛼𝜇𝜇 + (1 − 𝛼𝛼 ) 𝜇𝜇 = 𝜇𝜇 (b) 𝑉𝑉𝑉𝑉𝑉𝑉 ( 𝜇𝜇̂ ) = 𝛼𝛼 2 𝑉𝑉𝑉𝑉𝑉𝑉 ( 𝑋𝑋 1 ) + (1 − 𝛼𝛼 ) 2 𝑉𝑉𝑉𝑉𝑉𝑉 ( 𝑋𝑋 2 ) = 𝛼𝛼 2 𝜎𝜎 1 2 𝑛𝑛 1 + (1 − 𝛼𝛼 ) 2 𝜎𝜎 2 2 𝑛𝑛 2 = 𝛼𝛼 2 𝜎𝜎 1 2 𝑛𝑛 1 + 𝑉𝑉 (1 − 𝛼𝛼 ) 2 𝜎𝜎 1 2 𝑛𝑛 2 = 𝜎𝜎 1 2 𝛼𝛼 2 𝑛𝑛 2 + 𝑉𝑉 (1 −𝛼𝛼 ) 2 𝑛𝑛 1 𝑛𝑛 1 𝑛𝑛 2 SE( 𝜇𝜇̂ ) = 𝜎𝜎 1 𝛼𝛼 2 𝑛𝑛 2 + 𝑉𝑉 (1 −𝛼𝛼 ) 2 𝑛𝑛 1
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