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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#### Best Answer

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.