# A lower confidence sequence for the changing mean of non-negative right heavy-tailed observations with bounded mean

arXiv: https://arxiv.org/abs/2210.11133 · HTML: https://mlsanity.com/paper/2210.11133/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 14 references that are in the map.
- First posted on arXiv 20 Oct 2022.

## Builds on (most cited first, 5 of 5)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Catoni-style confidence sequences for heavy-tailed mean estimation](https://mlsanity.com/paper/2202.01250/) (2022)
- [Off-policy Confidence Sequences](https://mlsanity.com/paper/2102.09540/) (2021)
- [Anytime-valid sequential testing for elicitable functionals via supermartingales](https://mlsanity.com/paper/2204.05680/) (2022)

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

- [Risk-limiting Financial Audits via Weighted Sampling without Replacement](https://mlsanity.com/paper/2305.06884/) (2023)
- [Time-uniform confidence bands for the CDF under nonstationarity](https://mlsanity.com/paper/2302.14248/) (2023)

## Abstract (arXiv)

A confidence sequence (CS) is an anytime-valid sequential inference primitive which produces an adapted sequence of sets for a predictable parameter sequence with a time-uniform coverage guarantee. This work constructs a non-parametric non-asymptotic lower CS for the running average conditional expectation whose slack converges to zero given non-negative right heavy-tailed observations with bounded mean. Specifically, when the variance is finite the approach dominates the empirical Bernstein supermartingale of Howard et. al.; with infinite variance, can adapt to a known or unknown (1 + δ)-th moment bound; and can be efficiently approximated using a sublinear number of sufficient statistics. In certain cases this lower CS can be converted into a closed-interval CS whose width converges to zero, e.g., any bounded realization, or post contextual-bandit inference with bounded rewards and unbounded importance weights. A reference implementation and example simulations demonstrate the technique.