# Time-uniform confidence bands for the CDF under nonstationarity

arXiv: https://arxiv.org/abs/2302.14248 · HTML: https://mlsanity.com/paper/2302.14248/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 29 references that are in the map.
- First posted on arXiv 28 Feb 2023.

## Builds on (most cited first, 15 of 15)

- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Smoothed Analysis with Adaptive Adversaries](https://mlsanity.com/paper/2102.08446/) (2021)
- [Smoothed Online Learning is as Easy as Statistical Learning](https://mlsanity.com/paper/2202.04690/) (2022)
- [Smoothed Analysis of Online and Differentially Private Learning](https://mlsanity.com/paper/2006.10129/) (2020)
- [Universal Off-Policy Evaluation](https://mlsanity.com/paper/2104.12820/) (2021)
- [Anytime-valid off-policy inference for contextual bandits](https://mlsanity.com/paper/2210.10768/) (2022)
- [Off-Policy Risk Assessment in Contextual Bandits](https://mlsanity.com/paper/2104.08977/) (2021)
- [Sequential estimation of quantiles with applications to A/B-testing and best-arm identification](https://mlsanity.com/paper/1906.09712/) (2019)
- [Oracle-Efficient Online Learning for Beyond Worst-Case Adversaries](https://mlsanity.com/paper/2202.08549/) (2022)
- [Large Deviation Methods for Approximate Probabilistic Inference](https://mlsanity.com/paper/1301.7392/) (2013)
- [Risk-averse autonomous systems: A brief history and recent developments from the perspective of optimal control](https://mlsanity.com/paper/2109.08947/) (2021)
- [Rapid Regression Detection in Software Deployments through Sequential Testing](https://mlsanity.com/paper/2205.14762/) (2022)
- [Risk-Averse Multi-Armed Bandits with Unobserved Confounders: A Case Study in Emotion Regulation in Mobile Health](https://mlsanity.com/paper/2209.04356/) (2022)
- [A lower confidence sequence for the changing mean of non-negative right heavy-tailed observations with bounded mean](https://mlsanity.com/paper/2210.11133/) (2022)

## Built on by (most cited first, 2 of 2)

- [GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function](https://mlsanity.com/paper/2602.08096/) (2026)
- [Bet on Features: Anytime-Valid and Feature-Aware Auditing of Conditional Quantile Forecasters](https://mlsanity.com/paper/2607.11653/) (2026)

## Abstract (arXiv)

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.