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section10

# section10 - F ˆ θ f ˆ θ x = d dx F ˆ θ x = nx n-1 θ...

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OR 2700, Spring ’09 Section 10 Section 10 Apr 6 and Apr 7, 2009 Problem Suppose X 1 , . . . , X n constitute an iid sample from a uniform distribution on (0 , θ ), where θ is unknown. a. Compute the MLE of θ . b. Compute the cdf F ˆ θ of the estimator ˆ θ computed in part (a). c. Compute the pdf f ˆ θ of the estimator ˆ θ computed in part (a), and use it to obtain E ( ˆ θ ). Is ˆ θ an unbiased estimator for θ ? If not, construct an unbiased estimator for θ . d. Compute the standard error of the estimator ˆ θ computed in part (a). Solution a. The pdf of the uniform distribution on (0 , θ ) is f ( x ; θ ) = ( 1 0 < x < θ 0 otherwise so the joint pdf of X 1 , . . . , X n is f ( x 1 , . . . , x n ; θ ) = ( 1 n 0 < x i < θ for i = 1 , . . . , n 0 otherwise This density is maximized by choosing θ as small as possible. Since θ must be at least as large as all of the observed values x i , it follows that the smallest possible choice of θ is equal to max { x 1 , . . . , x n } . Hence the MLE of θ is ˆ θ = max { X 1 , . . . , X n } . b. We have F ˆ θ ( x ) = P ( ˆ θ x ) = P (max { X 1 , . . . , X n } ≤ x ) = P ( X 1 x, X 2 x, . . . , X n x ) = P ( X 1 x ) P ( X 2 x ) . . . P ( X n x ) = 0 x 0 ( x/θ ) n 0 < x < θ

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Unformatted text preview: F ˆ θ : f ˆ θ ( x ) = d dx F ˆ θ ( x ) = nx n-1 θ n , < x < θ . So the expected value of ˆ θ is E ( ˆ θ ) = Z θ x nx n-1 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 n-1 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 + 2-n 2 ( n + 1) 2 ¶ θ 2 and SE ˆ θ = s n n + 2-n 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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section10 - F ˆ θ f ˆ θ x = d dx F ˆ θ x = nx n-1 θ...

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