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Register a new userMinimax optimality of permutation tests
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Added by
Ilmun Kim
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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Permutation tests are widely used in statistics, providing a finite-sample guarantee on the type I error rate whenever the distribution of the samples under the null hypothesis is invariant to some rearrangement. Despite its increasing popularity and empirical success, theoretical properties of the permutation test, especially its power, have not been fully explored beyond simple cases. In this paper, we attempt to fill this gap by presenting a general non-asymptotic framework for analyzing the power of the permutation test. The utility of our proposed framework is illustrated in the context of two-sample and independence testing under both discrete and continuous settings. In each setting, we introduce permutation tests based on U-statistics and study their minimax performance. We also develop exponential concentration bounds for permuted U-statistics based on a novel coupling idea, which may be of independent interest. Building on these exponential bounds, we introduce permutation tests which are adaptive to unknown smoothness parameters without losing much power. The proposed framework is further illustrated using more sophisticated test statistics including weighted U-statistics for multinomial testing and Gaussian kernel-based statistics for density testing. Finally, we provide some simulation results that further justify the permutation approach.
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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.
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