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# slide6 - ECON321 Econometrics Lecture 6 Point Estimates...

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ECON321 : Econometrics Lecture 6 : Point Estimates Sasan Bakhtiari University of Maryland, College Park Summer 2007 Session II,

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Point Estimates A Random sample is a sequence X 1 , X 2 , . . . , X n of n random variables such that 1. X i ’s are independent, 2. X i ’s have identical probability ditributions. Example : I Age of n randomly chosen persons. I Profit of n randomly chosen firms. I ¯ X = 1 n n i =1 X i . I s 2 x = 1 n - 1 n i =1 ( X i - ¯ X ) 2 . ¯ X and s 2 X are the point estimates for μ x and σ 2 x .
Distribution of ¯ X I E [ ¯ X ] = E " 1 n n X i =1 X i # = 1 n n X i =1 E [ X i ] = 1 n n X i =1 μ x = μ x . I V ( ¯ X ) = V 1 n n X i =1 X i ! = 1 n 2 V n X i =1 X i ! = 1 n 2 n X i =1 σ 2 x = 1 n 2 n σ 2 x = σ 2 x n Notice that as n → ∞ , s 2 x 0.

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Special Cases I If all random variables X 1 , X 2 , . . . , X n are normal N ( μ x , σ 2 x ), then ¯ X N ( μ x , σ 2 x / n ) . I Central Limit Theorem - Consider a random sample of size n taken from an arbitrary distribution with mean μ x and variance σ 2 x . If n is sufficiently large, then the distribution of ¯ X
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