Data-Driven Rank Tests for Classes of Tail Alternatives

Abstract

Tail alternatives describe the occurrence of a nonconstant shift in the two-sample problem with a shift function increasing in the tail. The classes of shift functions can be built up using Legendre polynomials. It is important 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 are applied. Simulations show that the new data-driven rank tests work very well. Although other tests for detecting shift alternatives, such as Wilcoxon's test, may break down completely for important classes of tail alternatives, the new tests have high and stable power. The new tests also have higher power than data-driven rank tests for the unconstrained two-sample problem. Theoretical support is obtained by proving consistency of the new tests against very large classes of alternatives, including all common tail alternatives. A simple but accurate approximation of the null distribution makes application of the new tests easy.

KEY WORDS:

• Consistency
• Legendre polynomials
• Model selection
• Monte Carlo study
• Shift function