An adaptive test for the two-sample scale problem where the common quantile may be different from the median

In the usual two-sample scale problem it is assumed that the two populations have a common median. We consider the case where the common quantile may be other than a half. We investigate a quite general class, all members are based on UU-statistics where the minima and maxima of subsamples of various sizes are used. The asymptotic efficacies are investigated in detail. We construct an adaptive test where all statistics involved are suitably chosen. It is shown that the proposed adaptive test has good asymptotic and finite power properties.