4.2 Efficacy
The limit above can be computed without ever solving for \(n_1\) and \(n_2\), through a quantity that measures how sharply a statistic responds to a change in the parameter, relative to its own noise.
Definition 4.2.1 (Efficacy). Let \(T_n\) be a statistic with \(E_\theta (T_n) = \mu _n(\theta )\) and \(\operatorname {var}_\theta (T_n) = \sigma ^2_n(\theta )\), asymptotically normal under both \(H_0\) and local alternatives. Its efficacy is \[\mathrm {eff}(T) = \lim _{n\to \infty } \frac {\left [\mu _n'(\theta _0)\right ]^{2}}{n\,\sigma ^{2}_n(\theta _0)} .\]
The numerator is the rate at which the statistic moves when the parameter moves; the denominator is the statistic’s own variability. A test is good when the signal it responds to is large relative to the noise it carries, which is exactly what this ratio expresses.
Theorem 4.2.2 (Pitman). For two tests satisfying the regularity conditions above, \[\mathrm {ARE}\left (T_1,T_2\right ) = \frac {\mathrm {eff}(T_1)}{\mathrm {eff}(T_2)} .\]
Note 4.2.3. Theorem 4.2.2 is what makes the whole comparison tractable. Neither power function need be evaluated and no sample sizes need be solved for: it is enough to differentiate a mean and compute a variance under the null. Note also what has disappeared — the ARE depends on neither \(\alpha \) nor \(\beta \). Two tests stand in the same asymptotic relation to one another whatever level and power are demanded of them.
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