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ass4 - 03/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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03/2011 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Assignment 4 Section A Test statistics, p-values & critical regions A1. Let X 1 and X 2 be i.i.d. exponential random variables with mean θ . It is known that S = X 1 + X 2 has a gamma distribution with the density function f ( s ) = θ - 2 s e - s/θ , s > 0 . Define a region C k = { ( x 1 , x 2 ) : x 1 + x 2 > k } for some positive constant k . (a) Suppose the test with critical region C k is used to test H 0 : θ = 1 against H 1 : θ = 2. (i) Show that the test is equivalent to a likelihood ratio test. (ii) Show that the type I error probability of the test is (1 + k ) e - k . (iii) Find the type II error probability of the test. (iv) Show that one should choose k 4 . 744 in order to have a size 5% test. (v) Apply the size 5% test in (iv) to the observed data ( X 1 , X 2 ) = (2 . 3 , 3 . 0). What is your conclusion? Calculate the p-value of the test, and show how you would reach the above conclusion based on the p-value. (b) Suppose the test with critical region C k is used to test H 0 : θ 1 against H 1 : θ > 1. (i) Find the power function of the test and show that it is an increasing function in θ . (ii) Deduce from (i) the size of the test. (iii) Suppose that the test statistic S = X 1 + X 2 is observed to be s . Deduce from (i) and (ii) the p-value pv ( s ) associated with the observed value s . Show that pv ( s ) is a decreasing function in s . A2. Let U be uniformly distributed over [ θ - 1 , θ + 1]. To test H 0 : θ = 0 against H 1 : θ = 1, a test is proposed which rejects H 0 if U > 1 and accepts H 0 otherwise. (a) Find the type I error probability of the test. 1
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(b) Find the type II error probability of the test. (c) Find the size of the test. (d) Find the power function of the test. (e) Find the likelihood ratio of the hypotheses H 0 and H 1 if U is observed in the interval (i) ( - 1 , 0), (ii) [0 , 1], (iii) (1 , 2). A3. Let X 1 , . . . , X 10 be i.i.d. Bernoulli random variables with success probability θ , and ¯ X = 10 i =1 X i / 10. Define regions C 1 = { ¯ X > 0 . 8 } and C 2 = { ¯ X > 0 . 9 } ∪ { ¯ X < 0 . 1 } . (a) Find the power function of the critical region C 1 . (b) The critical region C 1 is used to test H 0 : θ 1 2 against H 1 : θ > 1 2 . Calculate the size of the test. (c) The critical region C 1 is used to test H 0 : θ 1 3 , 2 3 against H 1 : θ 6∈ 1 3 , 2 3 . Calculate the size of the test. (d) The critical region C 2 is used to test H 0 : θ 1 3 , 2 3 against H 1 : θ 6∈ 1 3 , 2 3 . (i) Find the power function of C 2 and the size of the above test. (ii) The critical region C 2 can be expressed in the form { T > k } for some test statistic T and critical value k . Specify T and k . (iii) Suppose that ¯ X is observed to be 0.5. Calculate the observed value t of T , and determine the p-value pv ( t ) sup H 0 P ( T > t | θ ). 2
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Section B Hypotheses & likelihood ratios In this section, you are required to (i) specify the null and alternative hypotheses in terms of unknown parameters, and (ii) calculate the likelihood ratio for the hypotheses for the following problems.
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