quiz4AS-f07 - F(y) = 1-θ y 0<y<1 = 0 elsewhere Where...

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ECON 506-M Quiz4-A, November 29, 2007 NAME: 1. y i are observations from a normal distribution with mean = 10 and variance = 4. The following is an estimator of the population mean: T = .7y 1 + .3y 2 , Find the bias and mean square error of the estimator. Solution: E (T) = E (.7y 1 + .3y 2 ) = 10, unbiased. V (T3) = 0.58 * 4 = 2.32, MSE = 2.32 2. Write the log likelihood function for a sample of m = 2 observations, Y 1 2 from a normal distribution. Solution: L = f (Y 1 =y 1 , Y 2 =y 1 ) = f (Y 1 =y 1 ) f (Y 2 =y 2 ) = = (1/ σ sqrt (2 π )) exp [- (1/2 σ 2 )(y 1 - μ ) 2 ] * (1/ sqrt (2 )) exp [- (1/2 σ 2 )(y 2 - μ ) 2 ]= = (1/ sqrt (2 )) 2 exp {- (1/2 σ 2 )[(y 1 - μ ) 2 + (y 2 - μ ) 2 ]} ln L = -2 ln σ + 2 ln sqrt (2 π ) - (1/2 σ 2 )[(y 1 - μ ) 2 + (y 2 - μ ) 2 ] 3. A log likelihood function for a certain distribution is the following. Find the maximum likelihood estimator of the parameter θ . Ln L = 8 ln θ - 3 θ - 5.0 Solution: Ln L = 8 ln θ - 3 θ - 5.0 d (ln L) / d θ = 8 θ -1 - 3 = 0 θ = 8/3 4. Let n Y Y Y , , , 2 1 denote a random sample from the probability density function
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Unformatted text preview: F(y) = 1-θ y 0<y<1 = 0 elsewhere Where . a. Find the method of moment estimator of . Solution: 1 ) ( 1 + = = ∫ dy y Y E . Therefore: Y Y Y-= ⇒ + = 1 ˆ 1 b. Show that Y is a consistent estimator of 1 + . Solution: 1 ) ( ) ( + = = θ Y E Y E ∞ → = ⇒ = n Y V n Y V Y V as ) ( lim ) ( ) ( 5. Y i are independent, identically distributed from a normal distribution. In our sample we find n = 20; 9 = Y , S 2 =43.158 a. Construct a 95% confidence interval for the mean. Solution: S =6.57. 95% CI is: n S t Y 19 025 . ± 9 ± 2.093 (6.57/√20) 9 ± 3.0748 [5.93, 12.07] b. Construct a 90% confidence interval for σ 2 . Solution: 117 . 10 , 1431 . 30 2 95 . 2 05 . = = χ 90% CI is [ ] 117 . 10 ) 158 . 43 )( 19 ( , 1435 . 30 ) 158 . 43 )( 19 ( [27.2, 81.05] 2...
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This note was uploaded on 04/13/2011 for the course ECON 506 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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quiz4AS-f07 - F(y) = 1-θ y 0<y<1 = 0 elsewhere Where...

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