Unformatted text preview: /I ( θ ). 10 . 4 a. Write ∑ X i Y i ∑ X 2 i = ∑ X i ( X i + ± i ) ∑ X 2 i = 1 + ∑ X i ± i ∑ X 2 i . From normality and independence E X i ± i = 0 , Var X i ± i = σ 2 ( μ 2 + τ 2 ) , E X 2 i = μ 2 + τ 2 , Var X 2 i = 2 τ 2 (2 μ 2 + τ 2 ) , and Cov( X i ,X i ± i ) = 0. Applying the formulas of Example 5.5.27, the asymptotic mean and variance are E ±∑ X i Y i ∑ X 2 i ² ≈ 1 and Var ±∑ X i Y i ∑ X 2 i ² ≈ nσ 2 ( μ 2 + τ 2 ) [ n ( μ 2 + τ 2 )] 2 = σ 2 n ( μ 2 + τ 2 ) b. ∑ Y i ∑ X i = β + ∑ ± i ∑ X i with approximate mean β and variance σ 2 / ( nμ 2 )....
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 Spring '12
 Dr.Hackney
 Statistics, Normal Distribution, Standard Deviation, Variance, Probability theory, probability density function, Estimation theory

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