Suppose we have a data set of n-observations where the extent of dependence between them is poorly understood. We assume we have an estimator that is squareroot-consistent for a particular estimand, and the dependence structure is weak enough so that the standardized estimator is asymptotically normally distributed. Our goal is to estimate the asymptotic variance of the standardized estimator so that we can construct a Wald-type confidence interval for the estimate. In this paper we present an approach that allows us to learn this asymptotic variance from a sequence of influence function based candidate variance estimators. We focus on time dependence, but the method we propose generalizes to data with arbitrary dependence structure. We show our approach is theoretically consistent under appropriate conditions, and evaluate its practical performance with a simulation study, which shows our method compares favorably with various existing subsampling and bootstrap approaches. We also include a real-world data analysis, estimating an average treatment effect (and a confidence interval) of ventilation rate on illness absence for a classroom observed over time.
Davies, Molly M. and van der Laan, Mark J., "Sieve Plateau Variance Estimators: A New Approach to Confidence Interval Estimation for Dependent Data" (May 2014). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 322.