## Abstract

Given independent samples from P andQ, two-sample permutation tests allow one to construct exact level tests when the null hypothesis is P = Q. On the other hand, when comparing or testing particular parameters θ of P and Q, such as their means or medians, permutation tests need not be level α, or even approximately level α in large samples. Under very weak assumptions for comparing estimators, we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability α in finite samples when the underlying distributions are identical. The ideas are broadly applicable and special attention is given to the k-sample problem of comparing general parameters, whereby a permutation test is constructed which is exact level α under the hypothesis of identical distributions, but has asymptotic rejection probability α under the more general null hypothesis of equality of parameters. A Monte Carlo simulation study is performed as well. A quite general theory is possible based on a coupling construction, as well as a key contiguity argument for the multinomial and multivariate hypergeometric distributions.

Original language | English (US) |
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Pages (from-to) | 484-507 |

Number of pages | 24 |

Journal | Annals of Statistics |

Volume | 41 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2013 |

Externally published | Yes |

## Keywords

- Behrens-Fisher problem
- Coupling
- Permutation test

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty