section10 - F : f ( x ) = d dx F ( x ) = nx n-1 n ,...

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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 ), 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 ) 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 1 θ x c. We compute f ˆ θ by taking the derivative of
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Unformatted text preview: F : f ( x ) = d dx F ( x ) = nx n-1 n , &lt; x &lt; . So the expected value of is E ( ) = Z x nx n-1 n dx = n ( n + 1) n x n +1 = n n + 1 . 1 OR 2700, Spring 09 Section 10 We see that E ( ) 6 = , which means that is a biased estimator for . Note that E n + 1 n = n + 1 n E ( ) = n + 1 n n n + 1 = , so n +1 n max { X 1 ,...,X n } is an unbiased estimator for . d. We compute E ( 2 ) = Z x 2 nx n-1 n dx = n ( n + 2) n x n +2 = n n + 2 2 . Therefore, Var( ) = n n + 2 2- n n + 1 2 = n n + 2-n 2 ( n + 1) 2 2 and SE = s n n + 2-n 2 ( n + 1) 2 . In order to estimate the standard error from the data, we may replace with . 2...
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This note was uploaded on 09/06/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell University (Engineering School).

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section10 - F : f ( x ) = d dx F ( x ) = nx n-1 n ,...

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