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Unformatted text preview: 2 ˆ p . 2 1 ) 2 1 ( ) ˆ ( 2 + + = + + = n np n Y E p E 2 2 1 2 ) 2 ( 1 2 1 += + ++ =+ + = n p n n p np p n np BIAS 1 b) (10) Derive ) ˆ ( 2 p MSE ( 29 2 2 2 ) 1 ( ) 2 1 ( ) ˆ ( += + + = n p np n Y V p V 2 2 2 ) 2 ( ) 2 1 ( ) 2 ( ) 1 ( ++ += n p n p np MSE 4. The Pareto distribution is frequently used as a model in study of incomes and has density function is given by: elsewhere 1 , 1 ) ; ( ) 1 ( = ≤ = +θ x x x f Suppose we have a random sample X 1 , X 2 , …, X n from this distribution. a) (12) Find the maximum likelihood estimator of . ( 29 ∑ ∑ += += = i n i i x n x L ln ) 1 ( ln * ln * ) 1 ( ln ln 1 ln / ln == ∂ ∂ ∑ i x n L ∑ = i x n ln ˆ b) (12) Find the method of moments estimator of . [ ] 1 ˆ 1 1 1 * * * ) ( 1 1 1 ) 1 (= ==∞= = =∞∞ +∫ ∫ x x x dx x dx x x x E 2...
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 Fall '08
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 Normal Distribution, ln xi

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