Testing Many and Possibly Singular Polynomial Constraints

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.