ch7_misc_solutions

ch7_misc_solutions - Miscellaneous Solutions and Comments...

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Miscellaneous Solutions and Comments on the Chapter 7 Exercises 7.4: For X 1 ,...,X n iid N ( θ, 1), the likelihood function may be found by setting μ = θ and σ 2 = 1 in the expression for the joint density given in Example 6.2.7 on page 277. This leads to L ( θ | x ) = c exp( - n x - θ ) 2 / 2) where c = (2 π ) - n/ 2 exp ± - n X i =1 ( x i - ¯ x ) 2 / 2 ! . The log-likelihood is thus ` ( θ ) = log c - n 2 x - θ ) 2 (1) which as a function of θ is a parabola which opens downward and achieves its maximum over all real values of θ at θ = ¯ x . In this problem we require θ 0. Let Θ = { θ : θ 0 } . The MLE is the value of θ Θ which maximizes ` ( θ ). If ¯ x 0, then the overall maximum at θ = ¯ x lies in Θ and is the MLE. However, if ¯ x < 0, then θ = ¯ x lies outside Θ. When ¯ x < 0 the log-likelihood is a decreasing function when restricted to Θ and the maximum in Θ is achieved at the endpoint θ = 0 which is the MLE in this case. (This is clear if you draw a picture of a parabola opening downward with its peak at ¯ x < 0.) In summary, the MLE is ˆ θ = ¯ x when ¯ x 0 and ˆ θ = 0 otherwise. You can reach the same conclusion without using the particular expression for ` ( θ ) given in (1) above. You just need to show that ` 0 ( θ ) = 0 when θ = ¯ x with ` 0 ( θ ) > 0 for θ < ¯ x and ` 0 ( θ ) < 0 for θ > ¯ x . Then if ¯ x < 0, you have
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ch7_misc_solutions - Miscellaneous Solutions and Comments...

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