Data-driven rank tests for classes of tail alternatives

Tail alternatives describe the occurrence of a nonconstant shift in the two -sample problem with a shift function increasing in the rail. The classes o f shift functions can be built up using Legendre polynomials. It is importa nt to choose the number of involved polynomials in the right way. Here this choice is based on the data, using a modification of the Schwarz selection rule. Given the data-driven choice of the model, appropriate rank tests ar e applied. Simulations show that the new data-driven rank tests work very w ell. Although other tests for detecting shift alternatives, such as Wilcoxo n's test, may break down completely for important classes of tail alternati ves, the new tests have high and stable power. The new tests also have high er power than data-driven rank tests for the unconstrained two-sample probl em. Theoretical support is obtained by proving consistency of the new tests against very large classes of alternatives, including all common tail alte rnatives. A simple but accurate approximation of the null distribution make s application of the new tests easy.