Small-sample efficiencies of rank tests

Nonparametric tests have been extensively investigated asymptotically for small signals and large numbers of samples. More realistic, though, in many engineering applications is the small-sample large-signal case, which has had little study because of its complexity. The problem of testing the hypothesis of probability density symmetry about a positive versus a negative value is investigated. The efficiency of the optimum rank and the efficiency of the Wilcoxon rank-sum test are found with respect to the optimum parametric test for normally distributed independent samples with large signal-to-noise ratios. Specifically, it is found that for the same signal-to-noise ratio and probability of error the optimum rank test requires at most 4/3 of the number of samples (or equivalently, 4/3 higher signal-to-noise ratio for the same number of samples) of the optimum parametric test; the Wilcoxon nonparametric test requires at most a factor of
2/\sqrt{2}
more. Thus, the efficiency of the Wilcoxon nonparametric test is very close to that of the optimum rank test for normal alternatives, although neither are as close to the efficiency of the optimum parametric test as in the large-sample small-signal problem (where, as is well known, the asymptotic relative efficiencies are
\pi/3
and 1).