In this chapter, we consider the asymptotic efficiency of tests which requires knowledge of the distribution of the test statistics under both the null and the alternative hypotheses. In the usual cases such as for the sign and the Wilcoxon tests, the calculations are straightforward. However, the calculations become more complicated in the multi-sample situations and for these, we appeal to Le Cam’s lemmas. This is illustrated in the case of test statistics involving both the Spearman and Hamming distances. The smooth embedding approach is useful in that for a given problem, it leads to test statistics whose power function can be determined. The latter is then assessed against the power function of the optimal test statistic derived from Hoeffding’s formula for any given underlying distribution of the data.
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