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Exercise 16.7.8. Suppose X1 .... ( Xn are iid N ((8) .0- ( ? ) ). (16.117) for -1 0. The problem is to find the likelihood ratio test of Ha: p =0,0' > 0 versus HA : p / 0,0' > 0. (16.118) Set T1 = [(X/ + Y?) and T2 = EXY,, the sufficient statistics. (a) Show that the MLE of o' is Ti/(2n) under Ho. (b) Under HA, let U; = (X; + Y;)/V/2 and V, = (X; - Y)/V2. Show that Up and V, are independent, U; ~ N(0, ) and Vi ~ N(0,02), 16.7. Exercises 281 where the = o'(1 + p) and 02 =o'(1 -p). (c) Find the MLEs of 6 and 62 in terms of the Uf's and V/'s. (d) Find the MLEs of of and p in terms of Ti and T2. (e) Use parts (a) and (d) to derive the form of the likelihood ratio test. Show that it is equivalent to rejecting Ho when 2(T2/T )2 > c