Rae Yu (余蕊琪)

This paper presents new uniform Gaussian strong approximations for empirical processes indexed by classes of functions based on \(d\)-variate random vectors (\(d \geq1 \)). First, a uniform Gaussian strong approximation is established for general empirical processes indexed by possibly Lipschitz functions, improving on previous results in the literature. In the setting considered by [Rio (1994)], and if the function class is Lipschitzian, our result improves the approximation rate \(n^{-1/(2d)}\)to \(n^{-1/\max\{d,2\}}\), up to a \(polylog(n)\)term, where \(n\)denotes the sample size. Remarkably, we establish a valid uniform Gaussian strong approximation at the rate \(n^{-1/2}\log n\)for \(d=2\), which was previously known to be valid only for univariate (\(d=1\)) empirical processes via the celebrated Hungarian construction [Komlós, Major, Tusnády (1975)]. Second, a uniform Gaussian strong approximation is established for multiplicative separable empirical processes indexed by possibly Lipschitz functions, which addresses some outstanding problems in the literature [Chernozhukov, Chetverikov, Kato (2014)]. Finally, two other uniform Gaussian strong approximation results are presented when the function class is a sequence of Haar basis based on quasi-uniform partitions. Applications to nonparametric density and regression estimation are discussed.

Uncertainty quantification in causal inference settings with random network interference is a challenging open problem. We study the large sample distributional properties of the classical difference-in-means Hajek treatment effect estimator, and propose a robust inference procedure for the (conditional) direct average treatment effect, allowing for cross-unit interference in both the outcome and treatment equations. Leveraging ideas from statistical physics, we introduce a novel Ising model capturing interference in the treatment assignment, and then obtain three main results. First, we establish a Berry-Esseen distributional approximation pointwise in the degree of interference generated by the Ising model. Our distributional approximation recovers known results in the literature under no-interference in treatment assignment, and also highlights a fundamental fragility of inference procedures developed using such a pointwise approximation. Second, we establish a uniform distributional approximation for the Hajek estimator, and develop robust inference procedures that remain valid regardless of the unknown degree of interference in the Ising model. Third, we propose a novel resampling method for implementation of robust inference procedure. A key technical innovation underlying our work is a new \textitDe-Finetti Machine that facilitates conditional i.i.d. Gaussianization, a technique that may be of independent interest in other settings.