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Unformatted text preview: log 2 X ̄ and this clearly differs from the previous t statistic. ∙ Both converge in distribution to Normal 0,1 under the null, and they have exactly the sample local power function. 68 ∙ An uncomfortable fact about largesample inference is that there are many different statistics that are asymptotically equivalent in the sense that the have the same local power function. One could conceivably reject H using one of the statistics but not the other. ∙ At a minimum, the asymptotic pvalues will usually differ. Suppose with n 400 we obtain x ̄ .8. and so ̂ exp − .8 ≈ .45. Rounding to two decimal places, 20 exp − .8 − .5 exp − .8 .8 ≈ − 2.52 20 .8 − log 2 .8 ≈ 2.39 69 (Note that if the alternative is H 1 : .5 this is equivalent to H 1 : log 2 , which is why the signs are different.) ∙ So, the asymptotic pvalues, gotten in both cases from the Normal 0,1 distribution, are different. (The onesided pvalues are about .0059 in the first case and about .0084 in the second case.) ∙ In practice, this has not been viewed as much of a problem, as there is often a “natural” way to compute the statistic. 70 ∙ The key to testing a single hypothesis using asymptotic methods is to just remember the general formula for a t statistic. If is a parameter and n ̂ − d → Normal 0, 2 , so that 2 is the asymptotic variance of n ̂ − (which depends on the parameter vector ), then the asymptotic standard error of ̂ is se ̂ 2 ̂ n ≡ ̂ n 71 ∙ The asymptotic t statistic for testing H : is ̂ − se ̂ n ̂ − ̂ , and this has a limiting standard normal distribution under H . Local power analysis can also be carried out. 72 Testing Multiple Restrictions ∙ Let be a p 1 vector in the open set Θ ⊂ p , and let c : Θ → q be a continuously differentiable function. Assume the q ≤ p restrictions to be tested are H : c ∙ For example, suppose p 3 and we want to test H : 1 2 1, 3 so there are q 2 restrictions. Then c c 1 c 2 1 2 − 1 3 73 ∙ Given n ̂ n − n d → Normal , V we use the delta method to obtain the asymptotic variance of n c ̂ n under the null hypothesis c : Avar n c ̂ n C VC ′ and the Wald test statistic is n c ̂ n ′ C ̂ n VC ̂ n ′ − 1 n c ̂ n c ̂ n ′ C ̂ n VC ̂ n ′ / n − 1 c ̂ n 74...
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 Fall '12
 Jeff
 Normal Distribution, Statistical hypothesis testing, Tn, local power

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