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7.9 - Further Details on Exercise 7.9 = X(n so that nxn1 n...

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Further Details on Exercise 7.9 ˆ θ = X ( n ) so that E ˆ θ = Z xf X ( n ) ( x ) dx = Z θ 0 x nx n - 1 θ n dx = n θ n Z θ 0 x n dx = n θ n θ n +1 n + 1 = n n + 1 θ E ˆ θ 2 = Z x 2 f X ( n ) ( x ) dx = Z θ 0 x 2 nx n - 1 θ n dx = n θ n Z θ 0 x n +1 dx = n θ n θ n +2 n + 2 = n n + 2 θ 2 Var( ˆ θ ) = n n + 2 θ 2 - n n + 1 θ 2 = 2 ( n + 2) ( n + 1) 2 . MSE = Bias 2 + Var = - θ n + 1 2 + 2 ( n + 2) ( n + 1) 2 = 2 θ 2 ( n + 1) ( n + 2)
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