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411_hw04_soln

411_hw04_soln - Stat 411 – Homework 04 Solutions 1...

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Unformatted text preview: Stat 411 – Homework 04 Solutions 1. Problem 6.2.7 in the text . The PDF for the Gamma (4 ,θ ) distribution is f θ ( x ) = 1 6 θ 4 x 3 e- x/θ , x > , θ > . (a) For the Fisher information, we first need second derivative of log-PDF: ∂ 2 ∂θ 2 log f θ ( x ) = ∂ 2 ∂θ 2 h const- 4 log θ- x θ i = 4 θ 2- 2 x θ 3 . If we recall that the expected value of a Gamma ( α,β ) random variable is αβ (see middle of p. 151 in the text), then I ( θ ) =- E θ h ∂ 2 ∂θ 2 log f θ ( X ) i = E θ (2 X ) θ 3- 4 θ 2 = 2 · 4 θ 2- 4 θ 2 = 4 θ 2 . (b) If X 1 ,...,X n iid ∼ Gamma (4 ,θ ), then the MLE is found by maximizing the log- likelihood: ‘ ( θ ) = log L ( θ ) = const- 4 n log θ- n x/θ. Setting the derivative equal to zero and solving for θ gives: ∂ ∂θ ‘ ( θ ) =- 4 n θ + n x θ 2 set = 0 ⇐⇒ ˆ θ = x 4 . If we recall that the variance of a Gamma ( α,β ) random variable is αβ 2 (see middle of p. 151 in the text), then V θ ( ˆ θ ) = V θ ( X 1 ) 16 n = 4 θ 2...
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411_hw04_soln - Stat 411 – Homework 04 Solutions 1...

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