Asymptotic Power and Efficiency of Lepage-Type Tests for the Treatment of Combined Location-Scale Alternatives

For the two-sample location and scale problem Lepage (1971) constructed a test that is based on a combination of the Wilcoxon test statistic and the Ansari-Bradley test statistic. We replace both components by arbitrary linear rank tests and obtain so-called Lepage-type tests that were introduced by Büning and Thadewald (2000). In the present paper we compute their asymptotic efficacies. The results of these calculations give rise to an idea how to construct adaptive tests based on the concept of Hogg (1974). We also include asymmetric densities in our study. It turns out that, for moderately skew densities, a combination of linear rank test statistics designed for symmetric densities is sufficient. Therefore, in our proposed adaptive test occur only tests designed for symmetric densities. For extremely skew densities the application of the combination of Savage-scores tests is suggested. A Monte Carlo study confirms the asymptotic results. Moreover, it shows that the adaptive test proposed is a serious competitor also for moderate sample sizes.