Online monotone density estimation and log-optimal calibration

TL;DR

This paper introduces online monotone density estimators, including an analogue of Grenander and an expert aggregation method, with theoretical guarantees and applications to sequential hypothesis testing calibration.

Contribution

It proposes novel online estimators for monotone densities, providing theoretical bounds and applying them to develop adaptive p-to-e calibrators for sequential testing.

Findings

Expected log-likelihood gap is $O(n^{1/3})$ in well-specified settings.

Pathwise regret for expert aggregation is $ ilde{O}( oot n)$ relative to the best offline estimator.

Proposed methods effectively calibrate p-to-e mappings in sequential hypothesis testing.

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.