ch7_misc_solutions

# ch7_misc_solutions - Miscellaneous Solutions and Comments...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

ch7_misc_solutions - Miscellaneous Solutions and Comments...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online