HW11_PS_Solution - ORIE 3500/5500 Fall Term 2008 Assignment...

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Assignment 11-Solution 1. The likelihood function is L ( θ ) = f X ( x 1 , .. . ,x n , θ ) = n Y i =1 f X i ( x i , θ ) = ± 1 (2 θ ) n if - θ x 1 , .. . ,x n θ 0 otherwise . Now - θ x 1 , .. . ,x n θ if and only if max 1 i n | x i | ≤ θ . So the MLE is ˆ θ = max 1 i n | X i | . Let F be its CDF. Then for 0 t θ F ( t ) = P ( ˆ θ t ) = P ( - t X i t i = 1 , .. . ,n ) = [ P ( - t X 1 t )] n = [ Z t - t (1 / 2 θ ) dt ] n = t n n . So the PDF is f ( t ) = nt n - 1 n for 0 t θ . So E ( ˆ θ ) = Z θ 0 nt n n dt = n n + 1 θ b θ ( ˆ θ ) = E ( ˆ θ ) - θ = - θ n + 1 . Now E ( ˆ θ 2 ) = Z θ 0 nt n +1 n dt = n n + 2 θ 2 , var ( ˆ θ ) = " n n + 2 - ² n n + 1 ³ 2 # θ 2 , and so MSE ( ˆ θ ) = var ( ˆ θ ) + b 2 θ ( ˆ θ ) = θ 2 " n n + 2 - ² n n + 1 ³ 2 + 1 ( n + 1) 2 # = θ 2 ´ n n + 2 - n - 1 n + 1 µ = 2 θ 2 ( n + 1)( n + 2) 2. The likelihood is L ( λ ) = p X ( x 1 , .. . ,x n , λ ) = n
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HW11_PS_Solution - ORIE 3500/5500 Fall Term 2008 Assignment...

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