Asymptotic Efficiency of Maesono Statistics for Testing of Symmetry

In an interesting paper Maesono introduced a new class of distribution-free statistics for testing of symmetry against shift alternative. The simplest of them coincides with the Wilcoxon statistic while the next is different but has the same Pitman efficiency. Maesono raised the problem of comparison between these two statistics on the basis of exact Bahadur efficiency. In this paper we calculate exact local Bahadur indices for all Maesono statistics and show when his statistics are better than the Wilcoxon statistic for sufficiently close alternatives.