Homework100B6

# Homework100B6 - ˆ β Problem 3 For the simple regression...

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STAT 100B HW 6 Due Friday 5pm Problem 1: For the simplest regression model y i = βx i + i , i = 1 , ..., n , suppose x i are fixed, and E( i ) = 0 and Var( i ) = σ 2 . (1) Let ˆ β = i w i y i , where w i may depend on x i . Identify w i so that ˆ β is unbiased and achieves the minimal variance. (2) Let ˆ β = i w i y i / i w i x i . Show that ˆ β is unbiased. Identify w i so that ˆ β achieves the minimal variance. Problem 2: For the simplest regression model y i = βx i + i , i = 1 , ..., n , suppose x i are fixed, and i N (0 , σ 2 ) independently. Then what is the distribution of the least squares estimate
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Unformatted text preview: ˆ β ? Problem 3: For the simple regression model y i = β + β 1 x i + ± i ,, i = 1 ,...,n , suppose x i are ﬁxed, and ± i ∼ N (0 ,σ 2 ) independently. (1) What is the distribution of the least squares estimate ˆ β 1 ? (2) Suppose we want to test H : β 1 = 0 versus H 1 : β 1 6 = 0. If we set the type I error to be α , then what is the decision rule? (3) Give a non-technical discussion on the issue of overﬁtting. 1...
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