610-Homework4 - Statistics 610 Homework 4 Moulinath...

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Statistics 610: Homework 4 Moulinath Banerjee University of Michigan Announcement: The Homework carries 120 points. Max possible score is 120. Due Monday, Nov 17. (1) Let X 1 , X 2 , . . . , X n be i.i.d. N ( μ, σ 2 ). Recall that we can construct a level 1 - α confidence interval for μ (when σ is unknown) by using a t –pivot that involves splitting α as β 1 + β 2 . Write down the interval in question and show the choice of β 1 = β 2 = α/ 2 is optimal in the sense that the interval then has smallest length. Let β ( μ ) denote the power function of the dual test for testing H 0 : μ = μ 0 in this model. Write down β ( μ ) analytically and show: (a) β ( μ 0 ) = 0 , ( b ) β ( μ ) converges to 1 as μ goes to -∞ or + . (c) β and β 0 are both symmetric functions about μ 0 . (d) Sketch a schematic graph of β . Hint: Conditioning on s , where s 2 is the unbiased estimate of σ 2 could help. (25) (2) In the above model, consider what is called the ”generalized likelihood ratio test” for testing μ = μ 0 against μ 6 = μ 0 , with σ unknown as before. This is given by: λ n = sup
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