4.3 The Efficacy of Rank Tests
We now compute the efficacies of the three procedures that matter most, under the location-shift model in which the alternative distribution is \(F(x-\theta )\) with density \(f\).
Theorem 4.3.1 (Efficacies under location shift). Assume \(f\) is absolutely continuous with finite Fisher information. Then \begin {align*} \text {$t$-test:}\qquad &\mathrm {eff}(t) = \frac {1}{\sigma ^{2}}, \\[4pt] \text {sign test:}\qquad &\mathrm {eff}(S) = 4f^{2}(0), \\[4pt] \text {Wilcoxon:}\qquad &\mathrm {eff}(W) = 12\left (\int ^{\infty }_{-\infty }f^{2}(x)\,dx\right )^{2}, \end {align*}
where \(\sigma ^{2}\) is the variance of \(F\).
Sketch for the Wilcoxon statistic. Take the two-sample statistic of Example 3.4.3 in the form \(U = P_n(X<Y)\) estimating \(\gamma (\theta ) = P(X < Y + \theta ) = \int F(y+\theta )f(y)\,dy\). Differentiating under the integral at \(\theta = 0\), \[\gamma '(0) = \int f(y)f(y)\,dy = \int f^{2}(y)\,dy .\] From the computation following Theorem 3.4.6, the null variance of \(U\) is \(\tfrac {1}{12}\) per observation in the relevant scaling. The efficacy is the squared derivative divided by that variance, giving \(\left (\int f^{2}\right )^{2}\big /\tfrac {1}{12} = 12\left (\int f^{2}\right )^{2}\). □
The \(\tfrac {1}{12}\) entering here is the variance of a uniform variable, arriving through the probability integral transform exactly as it did in Chapter 3. The efficiency of rank tests and their distribution-freeness have a common source.
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