# Online monotone density estimation and log-optimal calibration

arXiv: https://arxiv.org/abs/2602.08927 · HTML: https://mlsanity.com/paper/2602.08927/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 9 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Inductive Conformal Martingales for Change-Point Detection](https://mlsanity.com/paper/1706.03415/) (2017)
- [Sequentially valid tests for forecast calibration](https://mlsanity.com/paper/2109.11761/) (2021)

## Built on by (most cited first, 1 of 1)

- [Online LLM watermark detection via e-processes](https://mlsanity.com/paper/2602.14286/) (2026)

## Abstract (arXiv)

We study the problem of online monotone density estimation, where density estimators must be constructed in a predictable manner from sequentially observed data. We propose two online estimators: an online analogue of the classical Grenander estimator, and an expert aggregation estimator inspired by exponential weighting methods from the online learning literature. In the well-specified stochastic setting, where the underlying density is monotone, we show that the expected cumulative log-likelihood gap between the online estimators and the true density admits an O(n1/3) bound. We further establish a √nlogn pathwise regret bound for the expert aggregation estimator relative to the best offline monotone estimator chosen in hindsight, under minimal regularity assumptions on the observed sequence. As an application of independent interest, we show that the problem of constructing log-optimal p-to-e calibrators for sequential hypothesis testing can be formulated as an online monotone density estimation problem. We adapt the proposed estimators to build empirically adaptive p-to-e calibrators and establish their optimality. Numerical experiments illustrate the theoretical results.