4.6 Optimal Rank Tests
The efficacy formula also identifies which rank test is best for a given shape, which is the last question the chapter should answer.
Theorem 4.6.1 (Locally most powerful rank tests). Within the class of linear rank statistics of the form \(\sum a_n(R_i)\), the test maximising local power against location shift in \(F\) uses the scores \[a_n(i) = E\left [\varphi \left (U_{(i)}\right )\right ],\qquad \varphi (u) = -\frac {f'\left (F^{-1}(u)\right )}{f\left (F^{-1}(u)\right )} ,\] where \(U_{(i)}\) is the \(i\)th order statistic of a uniform sample.
Evaluating \(\varphi \) for particular densities recovers the familiar tests and explains their standing:
- the logistic density gives \(\varphi \) linear in \(u\), hence linear scores — the Wilcoxon test, which is therefore locally most powerful for logistic shift;
- the double exponential gives \(\varphi = \operatorname {sgn}\), hence the sign test, confirming Table 4.1;
- the normal gives \(\varphi = \Phi ^{-1}\), hence normal scores — the van der Waerden test, which attains \(\mathrm {ARE} = 1\) against the \(t\)-test under normality and is never worse than \(1\) against any distribution.
Remark 4.6.2. The last of these is remarkable and is not as widely known as it should be. The van der Waerden normal-scores test is fully distribution-free, yet its asymptotic efficiency relative to the \(t\)-test is at least \(1\) for every continuous \(F\), with equality only at the normal. There is, asymptotically, no price at all for the distribution-free guarantee — only the practical costs that the scores must be computed rather than read off, and that the small-sample behaviour is less well tabulated than Wilcoxon’s.
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