The exact power by defining n 1 n and then plugging

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Unformatted text preview: the exact power by defining n 1 n and then plugging this value of delta into the local power function. For example, if 1 .10 (say, a 10% increase in productivity) and n 400, then n .10 20 2, and the approximate power (with 1) is 2 − 1.65 .35 ≈ .64. 39 ∙ 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 ∙ 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 .05 .2 .4 .6 .8 1 local power-5 5 delta Local Power: Asymptotic Size = .05, sigma = 1 42 ∙ 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 ∙ 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 ∙ 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 – the mean in this case – but not the other parameters (the variance). 45 ∙ None of the preceding analysis relies on the null being of the form H : 0. In general, if the null is H : and H n 1 : n / n , the same sort of local power calculations work....
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