quiz4BS_f08 - 2 ˆ p 2 1 2 1 ˆ 2 = = n np n Y E p E 2 2 1...

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Econ 506, Section M NAME: __ ___ Ali Toossi __(solution)_________ Quiz 4B, December 4, 2008 Grade is out of 70 1. (12) Let Y 1 , Y 2 , …, Y n denote a random sample from the uniform distribution on the interval ( 1 , + θ ). Let 2 1 ˆ 1 - = Y . Show that 1 ˆ is a consistent estimator for . = - + = 2 1 2 1 ) ˆ ( 1 E n Y V V * 12 1 ) ( ) ˆ ( 1 = = Then, as n goes to infinity we have that bias is zero and variance goes to zero. 2 . The state of Illinois wishes to establish a 95% confidence interval for the standard deviation of personal consumption expenditures in the state. They have a sample of 30 observations and find that: Σ (Y – Y ) 2 = 200. a) (4) Do you need to make any additional assumption? If so, what? Yes. We should assume that the population has normal distribution b) (10) Construct the confidence interval. s 2 = 200 / (n-1) = 200 / 29 = 6.9 16.0471 < [(29 * 6.9) / σ 2 ]< 45.72 4.374 < σ2 < 12.46 2.091 < σ < 3.5298 3. If Y has a binomial distribution with parameters n and p, then n Y p = 1 ˆ is an unbiased estimator of p. Another estimator is ) 2 ( ) 1 ( ˆ 2 + + = n Y p . a) (10) Derive the bias of

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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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quiz4BS_f08 - 2 ˆ p 2 1 2 1 ˆ 2 = = n np n Y E p E 2 2 1...

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