Anytime Valid Confidence Bands for Distribution Functions
Hultberg et al.
Confidence bands for the distribution function of a data stream that hold simultaneously over all times and all points, even when each observation depends arbitrarily on the past.
Source published · In the Daily Brief of 28 October 2026
Filed under Probability, Learning theory
Classical bands for an unknown distribution function, such as those from the Dvoretzky–Kiefer–Wolfowitz inequality, assume a fixed sample size and independent data. This paper builds bands that remain valid uniformly over time, so they can be monitored continuously and stopped at any moment, and over every point of the domain.
The authors first treat the time-averaged conditional distribution function of a stream in which each observation may depend on the past, and show that the width of the band is governed by the depth of a binary search tree. They then present this as an instance of a general framework for any indexed family of time-averaged distribution functions, and apply it to importance-weighted, right-censored and multivariate distribution functions.