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Assignment07-2010 - as ˆ θ n 1 = x 1 1 n 1 x 2 1 n 1 ...

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AEB 6571 - Econometric Methods I Assignment 7 Charles B. Moss October 15, 2010 1. Consider the our discussion on estimating the θ parameter from the Bernoulli density function. Extending to the scenario to n draws (from 2 draws) the probability density function for drawing k events (i.e., x k =1)is P [ k,n,θ ]= n k ! θ k (1 θ ) n - k . (1) Given this formulation, our estimator of θ was simply ˆ θ = k n . (2) SpeciFcally, S = k/ 1and T = k/ 2=( x 1 + x 2 ) / 2. Given this formula- tion, graph the MSE = E ± ˆ θ θ ² 2 for n =2 , 3 , 4. 2. Note that the estimators for n 1 and n 2 where n 2 = n 1 +1 can be written
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Unformatted text preview: as ˆ θ n 1 = x 1 1 n 1 + x 2 1 n 1 + · · · x n 1 1 n 1 + x n 2 ˆ θ n 2 = x 1 1 n 2 + x 2 1 n 2 + · · · x n 2 1 n 2 + x n 2 1 n 2 . (3) Demonstrate that MSE ( θ n 1 ) ≥ MSE ( θ n 2 ). 1 3. Prove the Central Limit Theorem using the characteristic function. Is this theorem applicable to the estimator of θ in Question 2? 4. Derive the Maximum Likelihood estimator for the θ parameter of the Bernoulli distribution. 2...
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Assignment07-2010 - as ˆ θ n 1 = x 1 1 n 1 x 2 1 n 1 ...

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