Anytime Valid Confidence Bands for Distribution Functions
Abstract
Nonparametric inference often requires characterizing an unknown function, such as a cumulative distribution function (CDF), across its entire domain rather than a specific point. We develop confidence bands for distribution functions of a data stream that are valid simultaneously over all times and all points, even when each observation may depend arbitrarily on the past. We begin by deriving such bands for the time-averaged conditional CDF and show how their widths are determined by the depth of a binary search tree. We then show that this result is an instance of a framework through which one can construct valid confidence bands uniformly over time and over an arbitrary index set for any indexed family of time-averaged distribution functions. We illustrate the framework on importance-weighted and right-censored CDFs, as well as multivariate CDFs.
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BibTeX
@misc{hultberg2026anytime,
title = {{Anytime Valid Confidence Bands for Distribution Functions}},
author = {Bror Hultberg and Dave Zachariah and Antônio H. Ribeiro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36149},
eprint = {2610.36149},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36149},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Anytime Valid Confidence Bands for Distribution Functions
AU - Bror Hultberg
AU - Dave Zachariah
AU - Antônio H. Ribeiro
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36149
UR - https://arxiv.org/abs/2610.36149
AB - Nonparametric inference often requires characterizing an unknown function, such as a cumulative distribution function (CDF), across its entire domain rather than a specific point. We develop confidence bands for distribution functions of a data stream that are valid simultaneously over all times and all points, even when each observation may depend arbitrarily on the past. We begin by deriving such bands for the time-averaged conditional CDF and show how their widths are determined by the depth of a binary search tree. We then show that this result is an instance of a framework through which one can construct valid confidence bands uniformly over time and over an arbitrary index set for any indexed family of time-averaged distribution functions. We illustrate the framework on importance-weighted and right-censored CDFs, as well as multivariate CDFs.
ER -