Unformatted text preview: X ³ N & ;± 2 ± . Suppose that we have n random draws, X 1 , X 2 , ..., X n . (1) Since E ( X ) = 0 , we have ± 2 = E & X 2 ± . Obtain a sample analogue estimator of ± 2 . (2) Show that the maximum likelihood estimator of ± 2 is 1 n P n i =1 X 2 i . (3) Is 1 n P n i =1 X 2 i an unbiased estimator of ± 2 ? (4) Is 1 n P n i =1 X 2 i a consistent estimator of ± 2 ? 3. Suppose that the probability density function of a random variable X is given by p ( x ) = p ( x;² ) = ² x (1 ± ² ) 1 & x if x = 0 or 1 , and p ( x ) = p ( x;² ) = 0 otherwise. Suppose that we have n random draws, X 1 , X 2 , ..., X n . (1) Show that the expectation of X is ² . (2) Show that the maximum likelihood estimator of ² is b ² = 1 n P n i =1 X i . (3) Is b ² an unbiased estimator of ² ? (4) Is b ² a consistent estimator of ² ? 1...
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 Spring '08
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 Macroeconomics, Normal Distribution, Variance, Probability theory, probability density function

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