Question

Let X1,X be a random sample from an EXP(0) distribution (0 > 0) You will use the following facts for this question: Fact 1: If X EXP(0) then 2X/0~x(2). Fact 2: If V V, are a random sample from a x2(k) distribution then V V (nk) (a) Suppose that we wish to test Ho : 0 against H : 0 = 0, where 01 is specified and 0, > Oo. Show that the likelihood ratio statistic AE, O0,0)f(E)/ f (x;0,) is equal to exp (b) Using this expression, show that the most powerful critical region (re- jection region) of level a for testing Ho against Ha is where the quantile x2(m) is defined by P(W < x(m)) = a for W ~(m) (c) Explain why the test corresponding to the critical region C* described in part (b) is a uniformly most powerful (UMP) test of size a for against Ha 0> o testing Ho 0 (d) Suppose that we test Ho 0 against Ha 0> using the critical region C described in part (b). Show that the maximum probability of rejecting Ho when Ho is true is a

Answer 1

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