Online monotone density estimation and log-optimal calibration

2026-02-09

AI Summary

This study addresses the problem of online monotone density estimation and innovatively applies it to construct log-optimal p-to-e calibrators for sequential hypothesis testing. The authors propose two online estimators: an online variant of the classical Grenander estimator and a second method based on exponential weighting of expert aggregations. Under mild regularity conditions assuming the true density is monotone, the first estimator achieves a cumulative log-likelihood regret of $O(n^{1/3})$, while the second attains a pathwise regret bound of $O(\sqrt{n \log n})$ relative to the best offline monotone estimator. The theoretical analysis integrates tools from monotone function estimation, online learning, and sequential prediction. Numerical experiments corroborate the empirical effectiveness of the proposed methods.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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 $O(n^{1/3})$ bound. We further establish a $\sqrt{n\log{n}}$ 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.

Problem

• online monotone density estimation
• log-optimal calibration
• sequential hypothesis testing
• p-to-e calibrators

Innovation

• online monotone density estimation
• Grenander estimator
• expert aggregation
• log-optimal calibration
• pathwise regret bound