Efficiency of nonparametric tests for scale shift at an unknown time point

Recently Hsieh considered three nonparametric tests for the scale-change problem and showed them to be asymptotically as efficient as their parametric competitors. Here the class of so called sum-type statistics is studied which contains the statistics of Hsieh. It is proved that max-type statistics can be constructed that are at least as efficient in the sense of Bahadur as the sum-type statistics.