Online monotone density estimation and log-optimal calibration
This work addresses the challenge of constructing density estimators from sequential data in an online manner, with applications to log-optimal calibration for sequential hypothesis testing, representing an incremental advance in online learning methods.
The authors tackled the problem of online monotone density estimation by proposing two online estimators, showing that the expected cumulative log-likelihood gap is bounded by O(n^{1/3}) in a well-specified setting and achieving a √(n log n) pathwise regret bound for one estimator relative to the best offline choice.
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.