Recitation8Solutions

Recitation8Solutions - n ≤ x d x = θ-Z θ P(max X 1 X n...

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Steven Weber Dept. of ECE Drexel University ENGR 361: Statistical Analysis of Engineering Systems (Spring, 2008) Recitation 8 Solutions (Friday, May 23) Question 1. An rv X has CDF: F ( x ) = 0 , x < 0 ( x θ ) 2 , 0 x θ 1 , x > 1 . (1) A random sample ( X 1 , . . . , X n ) is gathered. The statistic ˆ θ = max { X 1 , . . . , X n } is proposed as a point estimator for the unknown parameter θ . Please do the following: Determine if the point estimator ˆ θ is biased ( E [ ˆ θ ] 6 = θ ) or unbiased ( E [ ˆ θ ] = θ ). If ˆ θ is biased, propose an unbiased point estimator. Prove your point estimator is unbiased. E [ ˆ θ ] = Z θ 0 P (max { X 1 , . . . , X n } > x ) d x = Z θ 0 (1 - P (max { X 1 , . . . , X
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Unformatted text preview: n } ≤ x )) d x = θ-Z θ P (max { X 1 , . . . , X n } ≤ x ) d x = θ-Z θ P ( X 1 ≤ x ) · · · P ( X n ≤ x )d x = θ-Z θ F ( x ) n d x = θ-Z θ ± x θ ² 2 n d x = θ-1 θ 2 n Z θ x 2 n d x = θ-x 2 n +1 (2 n + 1) θ 2 n ³ ³ ³ ³ θ = θ-θ 2 n + 1 = 2 n 2 n + 1 θ. (2) The estimator is biased. We propose: ˜ θ = 2 n +1 2 n ˆ θ . It is unbiased: E [ ˜ θ ] = 2 n + 1 2 n E [ ˆ θ ] = 2 n + 1 2 n 2 n 2 n + 1 θ = θ. (3) www.ece.drexel.edu/faculty/sweber 1 May 23, 2008...
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