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Dr. Hackney STA Solutions pg 125

Dr. Hackney STA Solutions pg 125 - 8-4 Solutions Manual for...

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8-4 Solutions Manual for Statistical Inference b. The LRT statistic is λ ( x ) = sup β (1 n ) e - Σ i x i sup β,γ ( γ n n )( i x i ) γ - 1 e - Σ i x γ i . The numerator is maximized at ˆ β 0 = ¯ x . For fixed γ , the denominator is maximized at ˆ β γ = i x γ i /n . Thus λ ( x ) = ¯ x - n e - n sup γ ( γ n / ˆ β n γ )( i x i ) γ - 1 e - Σ i x γ i / ˆ β γ = ¯ x - n sup γ ( γ n / ˆ β n γ )( i x i ) γ - 1 . The denominator cannot be maximized in closed form. Numeric maximization could be used to compute the statistic for observed data x . 8.8 a. We will first find the MLEs of a and θ . We have L ( a, θ | x ) = n i =1 1 2 πaθ e - ( x i - θ ) 2 / (2 ) , log L ( a, θ | x ) = n i =1 - 1 2 log(2 πaθ ) - 1 2 ( x i - θ ) 2 . Thus log L ∂a = n i =1 - 1 2 a + 1 2 θa 2 ( x i - θ ) 2 = - n 2 a + 1 2 θa 2 n i =1 ( x i - θ ) 2 set = 0 log L ∂θ = n i =1 - 1 2 θ + 1 2 2 ( x i - θ ) 2 + 1 ( x i - θ ) = - n 2 θ + 1 2 2 n i =1 ( x i - θ ) 2 + n ¯ x - set = 0 . We have to solve these two equations simultaneously to get MLEs of
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