We consider the problem of testing a null hypothesis defined by polynomial equality and inequality constraints on a statistical parameter. Testing such hypotheses can be challenging because the number of relevant constraints may be on the same order or even larger than the number of observed samples. Moreover, standard distributional approximations may be invalid due to singularities in the null hypothesis. We propose a general testing methodology that aims to circumvent these difficulties. The polynomials are estimated by incomplete U-statistics, and we derive critical values by Gaussian multiplier bootstrap. We prove that the bootstrap approximation of incomplete U-statistics is valid independently of the degeneracy of the kernel when the number of combinations used to compute the incomplete U-statistic is of the same order as the sample size. It follows that our test controls type I error over the whole parameter space and, in particular, it is valid at singularities. Furthermore, the bootstrap approximation covers high-dimensional settings making our testing strategy applicable for problems with many constraints. We study empirical size and power of the proposed tests in numerical experiments that assess the goodness-of-fit of latent tree models. Our implementation of the tests is available in an R package.

翻译：我们考虑了在统计参数上测试由多元不平等和不平等限制界定的无效假设的问题。测试这类假设可能具有挑战性,因为相关限制的数量可能与所观察到的样本数量相同,甚至更大。此外,标准的分布近似可能因无效假设中的奇数而无效。我们提议了一个旨在绕过这些困难的一般测试方法。多元模拟由不完全的U-统计学估算,我们从高斯乘数靴中得出关键值。我们证明,当用于计算不完整的U-统计模型的组合数量与样本大小相同时,不完整的U-统计学的靴套近似是有效的,独立于内核的退化性。因此,我们提出的用于计算不完整的U-统计模型的组合数量与样本大小相同,因此,我们的测试控制类型I在整个参数空间上出错,特别是它对于奇数是有效的。此外,靴杆近似覆盖了高维环境,使我们的测试战略适用于许多制约因素。我们研究了用于评估暗层树模型是否完善的数值实验中的拟议试验的经验规模和力量。