31 ∙ of course in the population the mean is either

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Unformatted text preview: 31 ∙ Of course, in the population the mean is either zero or some other value; saying the mean depends on the sample size makes little sense. We think of the population as being something fixed, and we sample from it. ∙ But Pitman drift allows us to approximate the power of alternatives that are not “too far” away from the null. The alternative “drifts” toward the null as the sample size grows. 32 ∙ As a technical matter, to use local alternatives we need extensions of the weak law of large numbers and the central limit theorem. These extensions exist under weak assumptions. ∙ If for each n ∈ 1,2,... , X ni : i 1,2,..., n is independent and identically distributed with mean n , then it is generally true that n − 1 ∑ i 1 n X ni − n X ̄ n − n p → 33 ∙ With further restrictions on higher moments – at a minimum, the second moments need to be finite – the CLT holds: X ̄ n − n n / n d → Normal 0,1 where n 2 Var X ni , i 1,2,..., n . ∙ Typically we would assume n 2 : n 1,2,... is a bounded sequence, if not actually constant. ∙ As usual, X ̄ n is standardized to have a zero mean and variance one to obtain a limiting standard normal distribution. 34 ∙ In the testing context where the null is H : 0 (or, actually, H : ≤ , we have n / n and n 2 2 for all n . Suppose we know 2 , so the t statistic is T n n X ̄ n n X ̄ n − n n n n X ̄ n − n n / n n X ̄ n − n d → Normal 0,1 Normal / ,1 35 ∙ The key point is that T n has a useful limiting distribution under the sequence of local alternatives – rather than diverging to , as under fixed alternatives. ∙ If we know , then we can find the asymptotic local power by computing P T c where T Normal / ,1 and c is the asymptotic critical value for a test with asymptotic size . 36 ∙ For example, suppose c 1.65. If / 1 then the local power is P T 1.65 P Z .65 ≈ .258 ∙ In general, for this example the local power function (which is always asymptotic in nature) for a test with asymptotic size .05 is 1 − 1.65 − / / − 1.65 Given that we (supposedly) know , we view this as a function of , and the local power clearly increases (to unity) as increases, for any given . 37 ∙ The local power function for 1: .05 .2 .4 .6 .8 1 local power 1 2 3 4 delta Local Power: Asymptotic Size = .05, sigma = 1 38 ∙ As a practical matter, given a mean value 1 that we would like to determine the power against, and given the sample size n , we can approximate the exact power by defining...
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31 ∙ Of course in the population the mean is either zero...

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