We study generalized Monte Carlo permutation tests under a non-uniform distribution on permutations. Focusing on the difference-in-means statistic, we introduce two scalar dispersion measures that quantify departures from complete randomization at the individual and pairwise levels. We show that if both dispersions vanish asymptotically, then the conditional permutation distribution converges to its Gaussian benchmark, the critical value stabilizes, and the test attains optimal Pitman local power. Conversely, if these dispersions fail to vanish, the permutation distribution does not self-average, the critical value need not stabilize, and optimal local power cannot in general be guaranteed. We further show that beyond the standard Pitman local model, suitably chosen non-uniform permutation distributions can strictly dominate the uniform distribution by exploiting nuisance structure in the data.

翻译：我们研究在排列非均匀分布下的广义蒙特卡洛排列检验。针对均值差统计量，我们引入两种标量色散度量，用于量化个体水平和成对水平上对完全随机化的偏离程度。研究表明，若两种色散渐近趋于零，则条件排列分布收敛至其高斯基准，临界值趋于稳定，且检验达到最优皮特曼局部功效。反之，若这些色散未能消失，则排列分布不自平均化，临界值未必稳定，且通常不能保证最优局部功效。我们进一步证明，在标准皮特曼局部模型之外，通过利用数据中的干扰结构，恰当选取的非均匀排列分布可严格优于均匀分布。