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Optimal allocation of Monte Carlo simulations to multiple hypothesis tests

机译：蒙特卡洛模拟对多个假设检验的最优分配

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Multiple hypothesis tests are often carried out in practice using p-value estimates obtained with bootstrap or permutation tests since the analytical p-values underlying all hypotheses are usually unknown. This article considers the allocation of a pre-specified total number of Monte Carlo simulations K is an element of N (i.e., permutations or draws from a bootstrap distribution) to a given number of m is an element of N hypotheses in order to approximate their p-values p is an element of[0,1]m in an optimal way, in the sense that the allocation minimises the total expected number of misclassified hypotheses. A misclassification occurs if a decision on a single hypothesis, obtained with an approximated p-value, differs from the one obtained if its p-value was known analytically. The contribution of this article is threefold: under the assumption that p is known and K is an element of R, and using a normal approximation of the Binomial distribution, the optimal real-valued allocation of K simulations to m hypotheses is derived when correcting for multiplicity with the Bonferroni correction, both when computing the p-value estimates with or without a pseudo-count. Computational subtleties arising in the former case will be discussed. Second, with the help of an algorithm based on simulated annealing, empirical evidence is given that the optimal integer allocation is likely of the same form as the optimal real-valued allocation, and that both seem to coincide asympotically. Third, an empirical study on simulated and real data demonstrates that a recently proposed sampling algorithm based on Thompson sampling asympotically mimics the optimal (real-valued) allocation when the p-values are unknown and thus estimated at runtime.
机译：在实践中，通常使用自举检验或置换检验获得的p值估算值进行多个假设检验，因为通常所有假设的分析p值都是未知的。本文考虑将预先指定的总数的蒙特卡罗模拟分配给K的N个元素（即，自举分布的置换或绘图）到给定​​的m个是N个假设的元素，以便近似它们的假设。 p值p以最佳方式是[0,1] m的元素，从某种意义上说，分配使误分类假设的总预期数量最小。如果使用近似p值获得的单个假设的决策与通过解析已知其p值获得的假设不同，则会发生分类错误。本文的贡献是三方面的：假设p是已知的，并且K是R的元素，并使用二项分布的正态近似，则在校正以下项时会得出K个模拟对m个假设的最优实值分配在计算带有或不带有伪计数的p值估算值时，都具有Bonferroni校正的多重性。将讨论在前一种情况下产生的计算细节。其次，借助基于模拟退火的算法，经验证据表明，最佳整数分配可能与最佳实值分配具有相同的形式，并且两者似乎是渐近一致的。第三，对模拟和真实数据的经验研究表明，最近提出的基于Thompson采样的采样算法会在p值未知并因此在运行时进行估计时，渐近地模拟最佳（实值）分配。

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