提出在线单调密度估计方法，实现高效概率校准。
Online monotone density estimation and log-optimal calibration
- 设计两种在线估计器：类格林纳德估计算法与专家加权聚合器。
- 在真实单调密度下，累积对数似然误差为 O(n^{1/3})。
- 适用于序列假设检验中的自适应 p-to-e 校准，适合统计学习研究者。
我们研究在线单调密度估计问题，即从顺序观测数据中以可预测方式构建密度估计器。提出两种在线估计器：一种是经典格林纳德估计的在线类比，另一种受在线学习中指数权重方法启发的专家聚合估计器。在底层密度为单调的充分设定下，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为 O(n^{1/3})。进一步在最小正则性假设下，建立专家聚合估计器相对于事后最优离线单调估计器的路径无关后悔界为 √(n log n)。作为独立兴趣的应用，我们表明构建序列假设检验中的对数最优 p-to-e 校准器问题可转化为在线单调密度估计问题。我们将所提估计器适配为经验自适应的 p-to-e 校准器，并证明其最优性。数值实验验证了理论结果。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。