quiz4AS_f08 - 1 ) ( 1 E n Y V V * 12 1 ) ( ) ( 1 = = Then,...

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Econ 506, Section M NAME: __ ___ Ali Toossi __(solution)_________ Quiz 4A, December 4, 2008 Grade is out of 70 1. 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 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 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 2. The Pareto distribution is frequently used as a model in study of incomes and has density function is given by: elsewhere 0 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 0 ln / ln = - = i x n L = i x n ln ˆ 1
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b) (12) Find the method of moments estimator of θ . [ ] 1 ˆ 1 1 1 * * * ) ( 1 1 1 ) 1 ( - = = - = - - = = = - - + - x x x dx x dx x x x E 3. (12) Let Y 1 , Y 2 , …, Y n denote a random sample from the uniform distribution on the interval ( 1 , + ). Let 2 1 ˆ 1 - = Y . Show that 1 ˆ is a consistent estimator for . = - + = 2 1 2
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Unformatted text preview: 1 ) ( 1 E n Y V V * 12 1 ) ( ) ( 1 = = Then, as n goes to infinity we have that bias is zero and variance goes to zero. 4 . 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: (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) / 2 ]< 45.72 4.374 < 2 < 12.46 2.091 < < 3.5298 2...
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quiz4AS_f08 - 1 ) ( 1 E n Y V V * 12 1 ) ( ) ( 1 = = Then,...

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