I have found the term "asymptotic power of a statistical test" only related to the Kolmogorov-Smirnov test (to be precise: asyptotic power = 1). What does this term acctually mean? In my opinion it should be someting like this: "if the alternative hypothesis is true, than for every significance level alpha there exists a sample size n that the selected test would reject the null hypothesis". Is "my" definition correct? According to "my defintion" the majority of classical tests (t-tset, ...) should have the asymptotic power 1, not only KS test. Am I right? ;)
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The definition above (a fixed alternative, sample size going to infinity) is more precisely related to the consistency (or not) of a hypothesis test. That is, a test is consistent against a fixed alternative if the power function approaches 1 at that alternative. Asymptotic power is something different. As Joris remarked, with asymptotic power the alternatives $\theta_n$ are changing, are converging to the null value $\theta_0$ (on the order of $\sqrt n$, say) while the sample size marches to infinity. Under some regularity conditions (for example, the test statistic has a monotone likelihood ratio, is asymptotically normal, has asymptotic variance $\tau$ continuous in $\theta$, yada yada yada) if $\sqrt n(\theta_n - \theta_0)$ goes to $\delta$ then the power function goes to $\Phi(\delta/\tau - z_\alpha)$, where $\Phi$ is the standard normal CDF. This last quantity is called the asymptotic power of just such a test. See Lehmann's $\underline{\mbox{Elements of Large Sample Theory}}$ for discussion and worked out examples. By the way, yes, the majority of classical tests are consistent. |
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As I understood it, the asymptotic power is the hypothetical power when the effect size goes to zero and the sample size to infinity. Basically it should be 0 or 1, indicating whether the test cannot or can distinguish an arbitrary small deviation from the null hypothesis when the sample size is sufficiently large. |
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Yes, you are right. I would only replace "there exists a sample size n that the selected test would reject the null hypothesis" with "for every e>0 there exists a sample size n_0 such that the probability to reject the null hypothesis is greater than 1-e for all n>n_0". |
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