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Unformatted text preview: F ˆ θ : f ˆ θ ( x ) = d dx F ˆ θ ( x ) = nx n1 θ n , < x < θ . So the expected value of ˆ θ is E ( ˆ θ ) = Z θ x nx n1 θ n dx = n ( n + 1) θ n x n +1 ﬂ ﬂ ﬂ ﬂ θ = n n + 1 θ . 1 OR 2700, Spring ’09 Section 10 We see that E ( ˆ θ ) 6 = θ , which means that ˆ θ is a biased estimator for θ . Note that E ± n + 1 n ˆ θ ¶ = n + 1 n E ( ˆ θ ) = n + 1 n · n n + 1 θ = θ , so n +1 n max { X 1 ,...,X n } is an unbiased estimator for θ . d. We compute E ( ˆ θ 2 ) = Z θ x 2 nx n1 θ n dx = n ( n + 2) θ n x n +2 ﬂ ﬂ ﬂ ﬂ θ = n n + 2 θ 2 . Therefore, Var( ˆ θ ) = n n + 2 θ 2± n n + 1 θ ¶ 2 = ± n n + 2n 2 ( n + 1) 2 ¶ θ 2 and SE ˆ θ = s n n + 2n 2 ( n + 1) 2 θ . In order to estimate the standard error from the data, we may replace θ with ˆ θ . 2...
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 Spring '05
 STAFF
 Normal Distribution, Bias of an estimator, Xn

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