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Dr. Hackney STA Solutions pg 102

# Dr. Hackney STA Solutions pg 102 - Second Edition 7-5 This...

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Second Edition 7-5 This is a linear function of θ that decreases for j < n/ 2 and increases for j > n/ 2. If n is even, 2 j - n = 0 if j = n/ 2. So the likelihood is constant between x ( n/ 2) and x (( n/ 2)+1) , and any value in this interval is the MLE. Usually the midpoint of this interval is taken as the MLE. If n is odd, the likelihood is minimized at ˆ θ = x (( n +1) / 2) . 7.15 a. The likelihood is L ( μ,λ | x ) = λ n/ 2 (2 π ) n Q i x i exp ± - λ 2 X i ( x i - μ ) 2 μ 2 x i ² . For ﬁxed λ , maximizing with respect to μ is equivalent to minimizing the sum in the expo- nential. d X i ( x i - μ ) 2 μ 2 x i = d X i (( x i ) - 1) 2 x i = - X i 2 (( x i ) - 1) x i x i μ 2 . Setting this equal to zero is equivalent to setting X i ³ x i μ - 1 ´ = 0 , and solving for μ yields ˆ μ n = ¯ x . Plugging in this ˆ μ n and maximizing with respect to λ amounts to maximizing an expression of the form λ n/ 2 e - λb . Simple calculus yields ˆ λ n = n 2 b where b = X i ( x i - ¯ x ) 2 x 2 x i . Finally, 2 b = X i x i ¯ x 2 - 2 X i 1 ¯ x + X i 1 x i = - n ¯ x + X i 1 x i = X i ³ 1 x i - 1 ¯ x ´ . b. This is the same as Exercise 6.27b.
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