## The Annals of Statistics

### Exact and asymptotically robust permutation tests

#### Abstract

Given independent samples from $P$ and $Q$, 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 $\theta$ of $P$ and $Q$, such as their means or medians, permutation tests need not be level $\alpha$, or even approximately level $\alpha$ 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 $\alpha$ 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 $\alpha$ under the hypothesis of identical distributions, but has asymptotic rejection probability $\alpha$ 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.

#### Article information

Source
Ann. Statist., Volume 41, Number 2 (2013), 484-507.

Dates
First available in Project Euclid: 16 April 2013

https://projecteuclid.org/euclid.aos/1366138199

Digital Object Identifier
doi:10.1214/13-AOS1090

Mathematical Reviews number (MathSciNet)
MR3099111

Zentralblatt MATH identifier
1267.62064

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G10: Hypothesis testing

#### Citation

Chung, EunYi; Romano, Joseph P. Exact and asymptotically robust permutation tests. Ann. Statist. 41 (2013), no. 2, 484--507. doi:10.1214/13-AOS1090. https://projecteuclid.org/euclid.aos/1366138199

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