  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.
