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411_hw03 - Newton’s method covered in early calculus...

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Stat 411 – Homework 03 Due: Friday 02/03 1. Problem 6.1.3 on pages 317–318. 2. Problem 6.1.6 on page 318. [Hint: Refer back to Problem 6.1.3(c).] 3. Problem 6.1.9 on page 318. [Hint: Refer back to Problem 6.1.3(a).] 4. Problem 6.1.11 on page 319. 5. Let θ be a scalar (one-dimensional) parameter and let ( θ ) denote the log-likelihood function. Then the MLE ˆ θ is a solution of the likelihood equation 0 ( θ ) = 0. It may happen that there is a unique solution to the likelihood equation, but there’s no formula for it (e.g., in a gamma shape parameter problem). In such cases, the solution must be found numerically. A powerful method for such problems is
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Unformatted text preview: Newton’s method , covered in early calculus courses. Describe how Newton’s method can be used to solve the likelihood equation. You may assume that ‘ ( θ ) is at least twice differentiable. 6. (Graduate Only) Consider independent samples from two normal populations: X 11 ,...,X 1 n 1 iid ∼ N ( θ 1 , 1) and X 21 ,...,X 2 n 2 iid ∼ N ( θ 2 , 1) , independent throughout. Suppose that θ 1 ≤ θ 2 . Find the MLE of ( θ 1 ,θ 2 ). You may find it helpful to sketch the parameter space Θ = { ( θ 1 ,θ 2 ) : θ 1 ≤ θ 2 } . 1...
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