### Abstract

Tail alternatives describe the frequent occurrence of a non-constant 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 rightly choose the number of polynomials involved. Here this choice is based on the data, using a modification of Schwarz's 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. While other tests for detecting shift alternatives as Wilcoxon's test may completely break down for important classes of tail alternatives, the new tests have high and stable power. The new tests have also 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.

Original language | Undefined |
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Pages (from-to) | 685-696 |

Number of pages | 12 |

Journal | Journal of the American Statistical Association |

Volume | 96 |

Issue number | 454 |

Publication status | Published - 2001 |

### Keywords

- EWI-12842
- MSC-62E25
- MSC-62G10
- MSC-62G20
- Model selection
- Legendre polynomials
- Shift function
- Consistency
- IR-62342
- Monte Carlo study

## Cite this

Albers, WW., Kallenberg, W. C. M., & Martini, F. (2001). Data driven rank tests for classes of tail alternatives.

*Journal of the American Statistical Association*,*96*(454), 685-696.