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，其中密度估计器必须根据顺序观测数据以可预测的方式构建。我们提出了两种在线估计器：经典Grenander估计器的在线模拟版本，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。在设定正确的随机场景下（即基础密度具有单调性），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存在$O(n^{1/3})$上界。进一步地，在对观测序列施加最小正则性假设的条件下，我们针对专家聚合估计器建立了相对于事后选择的最优离线单调估计器的$\sqrt{n\log{n}}$路径悔值界。作为具有独立价值的应用，我们证明序列假设检验中构建对数最优p-to-e校准器的问题可表述为在线单调密度估计问题。我们将所提出的估计器适配于构建经验自适应的p-to-e校准器，并证明其最优性。数值实验验证了理论结果。