Second Edition8-198.45 The verification of sizeαis the same computation as in Exercise 8.37a. Example 8.3.3 showsthat the power functionβm(θ) for each of these tests is an increasing function. So forθ > θ0,βm(θ)> βm(θ0) =α. Hence, the tests are all unbiased.8.47 a. This is very similar to the argument for Exercise 8.41.b. By an argument similar to part (a), this LRT rejectsH+0ifT+=¯X-¯Y-δS2p(1n+1m)≤ -tn+m-2,α.c. BecauseH0is the union ofH+0andH-0, by the IUT method of Theorem 8.3.23 the testthat rejectsH0if the tests in parts (a) and (b) both reject is a levelαtest ofH0. That is,the test rejectsH0ifT+≤ -tn+m-2,αandT-≥tn+m-2,α.d. Use Theorem 8.3.24. Consider parameter points withμX-μY=δandσ→0. For anyσ,P(T+≤ -tn+m-2,α) =α. The power of theT-test is computed from the noncentraltdistribution with noncentrality parameter|μx-μY-(-δ)|/[σ(1/n+ 1/m)] = 2δ/[σ(1/n+1/m)] which converges to∞asσ→0. Thus,P(T-≥tn+m-2,α)→1 asσ→0. By Theorem8.3.24, this IUT is a size
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