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39 we can use the same approach for two sided

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39
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We can use the same approach for two-sided alternatives. Now, when we write H n 1 : n / n , can be negative as well as positive. We get the same limiting distribution for T n : T n d T Normal / ,1 where can be negative or positive. 40
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Choose a 5% asymptotic size, so c 1.96. Then we need to compute P | T | 1.96 1 P 1.96 T 1.96 1 P 1.96 / T / 1.96 / 1 1.96 / 1.96 /  1.96 / 1.96 / The shape of the local power function is identical to that for the exact power function under normality. 41
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.05 .2 .4 .6 .8 1 local power -5 0 5 delta Local Power: Asymptotic Size = .05, sigma = 1 42
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Recall that the asymptotic distribution of the t statistic under the null hypothesis is the same when we replace with S n , the sample standard deviation. This continues to be true under local alternatives: T n n X ̄ n S n n X ̄ n n n n S n n X ̄ n n S n S n d Normal 0,1 Normal / ,1 43
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The derivation uses n X ̄ n n S n n X ̄ n n o p 1 d Normal 0,1 S n o p 1 A consequence is that for local power analysis, we can act as if S n is . This is not true for exact power analysis, where we move from the normal distribution (using ) to a t distribution (using S n ). 44
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When is unknown, the best we can do is estimate the local power function. For example, against H n 1 : n / n the estimator of the power function is ̂ n / S n 1.65 The same local power analysis goes through if we replace with n where n 0 and S n n o p 1 This is why we usually only model the drift for the parameters that we are testing under H 0 – the mean in this case – but not the other parameters (the variance). 45
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None of the preceding analysis relies on the null being of the form H 0 : 0. In general, if the null is H 0 : 0 and H n 1 : n 0 / n , the same sort of local power calculations work. Suppose that, in the earlier Poisson example, the null is H 0 : 2. The three statistics are T n 1 n X ̄ n 2 2 , T n 2 n X ̄ n 2 X ̄ n , T n 3 n X ̄ n 2 S n Not only do these all have limiting Normal 0,1 distributions under H 0 , they all have limiting Normal / 2 ,1 distributions under H n 1 . 46
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We can show this by using more general results. Let n be the mean of the random sample X ni : i 1,2,..., n and n 2 Var X ni . We take as given that the CLT holds, that is, n X ̄ n n n d Normal 0,1 .
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