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assign1 - α ² where χ 2 α is the α critical value for...

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PSTAT 120C: Assignment # 1 Due April 9, 2009 These problems will be due in lecture next Thursday. 1. Suppose that we have 9 independent observations from a normal distribution with standard deviation 10. We wish to test H 0 : μ = 150 vs. H A : μ 6 = 150 The GLRT with level α = 0 . 05 has the critical set C 1 = { ¯ x < 143 . 5 or ¯ x > 156 . 5 } . (a) Calculate the power of this test against the alternative μ = 151. (b) Calculate the power of this test against the alternative μ = 123. (c) Alternatively, consider the test with C 2 = { ¯ x < 144 . 5 or ¯ x > 160 . 9 } . i. Calculate the level of this second test. ii. Calculate the power against the alternatives μ = 151 and μ = 123. (d) Which of the two tests do you think is better? Why? 2. This problem is not due until April 16 (Exercise 10.105 on page 554 in WMS .) Let Y 1 ,...,Y n denote a random sample from a normal distribution with mean μ (unknown) and variance σ 2 . For testing H 0 : σ 2 = σ 2 0 against H A : σ 2 > σ 2 0 , show that the likelihood ratio test is equivalent to the χ 2 test with critical region C = ± S 2 ( n - 1) σ 2 0 > χ 2
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Unformatted text preview: α ² where χ 2 α is the α critical value for a χ 2 distribution with n-1 degrees of freedom. 3. Suppose that Y is a random sample of size 1 from a population with density f ( y | θ ) = ( θy θ-1 , ≤ y ≤ 1 , elsewhere. where θ > 0. (a) Sketch the power function of the test with the critical region Y > . 5. (Compute the power at a reasonable number of points and draw a curve that seems to ﬁt those values. Please be neat.) (b) Find the best test for testing H : θ = 1 versus H A : θ = 2 of size α . ( ? ) Show that you get the same test for any alternative where H A : θ = t and t > 1. 4. Show that if T is a suﬃcient statistic for estimating θ from the data X 1 ,...,X n then the Generalized Likelihood Ratio Test (GLRT) statistic for testing H : θ ∈ Ω versus H a : θ 6∈ Ω is a function of T . (Hint: use the factorization theorem.) 1...
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