# Time-uniform confidence bands for the CDF under nonstationarity

## TL;DR

This paper develops valid, time-uniform confidence bands for the cumulative distribution function (CDF) of a random variable in nonstationary settings, extending to importance-weighted cases for counterfactual analysis.

## Contribution

It introduces the first computationally feasible, always valid confidence bounds on the CDF under nonstationarity, with convergence guarantees and applicability to importance-weighted data.

## Key findings

- Provides time-uniform confidence bands valid under nonstationarity.
- Extends bounds to importance-weighted estimations for counterfactual distributions.
- Guarantees convergence in arbitrary data-dependent environments.

## Abstract

Estimation of the complete distribution of a random variable is a useful primitive for both manual and automated decision making. This problem has received extensive attention in the i.i.d. setting, but the arbitrary data dependent setting remains largely unaddressed. Consistent with known impossibility results, we present computationally felicitous time-uniform and value-uniform bounds on the CDF of the running averaged conditional distribution of a real-valued random variable which are always valid and sometimes trivial, along with an instance-dependent convergence guarantee. The importance-weighted extension is appropriate for estimating complete counterfactual distributions of rewards given controlled experimentation data exhaust, e.g., from an A/B test or a contextual bandit.