论文标题
使用不完整的U统计数据在可能不规则模型中测试许多限制
Testing Many Constraints in Possibly Irregular Models Using Incomplete U-Statistics
论文作者
论文摘要
我们考虑测试由统计参数的平等和不平等约束定义的零假设的问题。检验此类假设可能具有挑战性，因为相关约束的数量可能在相同的顺序上，甚至比观察到的样本数量大。此外，由于零假设中的不规则性，标准分布近似可能无效。我们提出了一种一般测试方法，旨在规避这些困难。约束是通过不完整的U统计量来估计的，我们通过高斯乘数引导程序得出临界值。我们表明，不完整的U统计量的自举近似对于内核有效，当用于计算不完整U统计的组合数与样本尺寸的顺序相同时，我们称之为混合变性。因此，即使在不规则设置中，我们的测试控制I型错误也是如此。此外，引导程序近似涵盖了高维设置，使我们的测试策略适用于具有许多约束的问题。当要测试的约束是u-ostable参数中的多项式时，该方法是适用的。作为应用程序，我们考虑了用于多元数据的潜在树模型的合适性测试。
We consider the problem of testing a null hypothesis defined by 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 irregularities in the null hypothesis. We propose a general testing methodology that aims to circumvent these difficulties. The constraints are estimated by incomplete U-statistics, and we derive critical values by Gaussian multiplier bootstrap. We show that the bootstrap approximation of incomplete U-statistics is valid for kernels that we call mixed degenerate 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 even in irregular settings. Furthermore, the bootstrap approximation covers high-dimensional settings making our testing strategy applicable for problems with many constraints. The methodology is applicable, in particular, when the constraints to be tested are polynomials in U-estimable parameters. As an application, we consider goodness-of-fit tests of latent tree models for multivariate data.
