Central limit theorems via Stein's method for randomized experiments under interference
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Controlling for interference through design and analysis can consume both engineering resources and statistical power, so it is of interest to understand the extent to which estimators and confidence intervals constructed under the SUTVA assumption are still valid in the presence of interference. Toward this end, Sävje et al. (2017) provide laws of large numbers for standard estimators of the average treatment effect under a limited form of interference characterized by an interference dependence graph. In this paper we link that view of interference to the dependency graph version of Stein's method. We prove a central limit theorem for a variant of the difference-in-means estimator if the o(n) restriction on average dependency degree in Sävje et al. (2017) is replaced by an o(n^1/4) constraint on the maximal dependency degree. We then provide a central limit theorem that can handle interference that exists between all pairs of units, provided the interference is approximately local. The asymptotic variance admits a decomposition into two terms: (a) the variance that is expected under no-interference and (b) the additional variance contributed by interference. The results arise as an application of two flavors of Stein's method: the dependency graph approach and the generalized perturbative approach.
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