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.

翻译：本文研究在线单调密度估计问题，要求从顺序观测数据中以可预测方式构建密度估计器。我们提出两种在线估计方法：经典Grenander估计器的在线版本，以及受在线学习文献中指数加权方法启发的专家聚合估计器。在设定良好的随机场景（即底层密度为单调函数）下，我们证明在线估计器与真实密度之间的期望累积对数似然差距存在$O(n^{1/3})$上界。进一步地，在观测序列的最小正则性假设下，我们建立了专家聚合估计器相对于事后最优离线单调估计器的$\sqrt{n\log{n}}$逐路径遗憾界。作为一项具有独立意义的应用，我们证明序贯假设检验中对数最优p-to-e校准器的构建问题可转化为在线单调密度估计问题。我们通过调整提出的估计器构建了经验自适应p-to-e校准器，并证明了其最优性。数值实验验证了理论结果。