When the target of inference is a real-valued function of probability parameters in the k-sample multinomial problem, variance estimation may be challenging. In small samples, methods like the nonparametric bootstrap or delta method may perform poorly. We propose a novel general method in this setting for computing exact p-values and confidence intervals which means that type I error rates are correctly bounded and confidence intervals have at least nominal coverage at all sample sizes. Our method is applicable to any real-valued function of multinomial probabilities, accommodating an arbitrary number of samples with varying category counts. We describe the method and provide an implementation of it in R, with some computational optimization to ensure broad applicability. Simulations demonstrate our method's ability to maintain correct coverage rates in settings where the nonparametric bootstrap fails.

翻译：当推断目标为k样本多项问题中概率参数的实值函数时，方差估计可能具有挑战性。在小样本情况下，非参数自助法或Delta法等方法的性能可能不佳。我们在此背景下提出了一种计算精确p值和置信区间的新颖通用方法，这意味着I类错误率被正确限定，且置信区间在所有样本量下至少具有名义覆盖率。我们的方法适用于多项概率的任何实值函数，可容纳具有不同类别数量的任意数量样本。我们描述了该方法并在R语言中提供了其实现，同时进行了一些计算优化以确保广泛的适用性。仿真实验证明了我们的方法在非参数自助法失效的场景中保持正确覆盖率的能力。