Confidence intervals for means under constrained dependence

TL;DR

This paper introduces a framework for constructing confidence intervals for the mean of dependent variables with known or constrained dependency graphs, ensuring asymptotic coverage and providing practical algorithms and software.

Contribution

It develops new methods for variance estimation and confidence interval construction under dependency constraints, including unknown graphs with topological and degree constraints.

Findings

Proves consistency of an oracle variance estimator.

Derives bounds for variance estimation with unknown dependency graphs.

Provides software implementation for practical use.

Abstract

We develop a general framework for conducting inference on the mean of dependent random variables given constraints on their dependency graph. We establish the consistency of an oracle variance estimator of the mean when the dependency graph is known, along with an associated central limit theorem. We derive an integer linear program for finding an upper bound for the estimated variance when the graph is unknown, but topological and degree-based constraints are available. We develop alternative bounds, including a closed-form bound, under an additional homoskedasticity assumption. We establish a basis for Wald-type confidence intervals for the mean that are guaranteed to have asymptotically conservative coverage. We apply the approach to inference from a social network link-tracing study and provide statistical software implementing the approach.