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# 07 - THE UNIVERSITY OF HONG KONG Department of Statistics...

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THE UNIVERSITY OF HONG KONG Department of Statistics and Actuarial Science STAT 1302 PROBABILITY AND STATISTICS II Tutorial 7 1. Let X 1 and X 2 be i.i.d. exponential random variables with mean θ . It is well known that S = X 1 + X 2 has a gamma distribution with the density function f ( s ) = s · ( e - s/θ ) 2 , s > 0 . Deﬁne a region C k = { ( x 1 ,x 2 ) : x 1 + x 2 > k } for some positive k . (a) Suppose that 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 has the form of 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. 1

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(b) Suppose the test with critical region C k is used to test H 0 : θ 1 against H 1 : θ > 1 . i. Show that the loglikelihood ratio of the hypotheses H 0 and H 1 is log Λ( H 0 ,H 1 ) = | X 1 + X 2 - 2 | - 2 | log[( X 1 + X 2 ) / 2] | ii. Find the power function of the test and show that it is monotonically increas- ing. iii. Deduce from (ii) the size of the test.
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07 - THE UNIVERSITY OF HONG KONG Department of Statistics...

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