The simultaneous estimation of multiple unknown parameters lies at heart of a broad class of important problems across science and technology. Currently, the state-of-the-art performance in the such problems is achieved by nonparametric empirical Bayes methods. However, these approaches still suffer from two major issues. First, they solve a frequentist problem but do so by following Bayesian reasoning, posing a philosophical dilemma that has contributed to somewhat uneasy attitudes toward empirical Bayes methodology. Second, their computation relies on certain density estimates that become extremely unreliable in some complex simultaneous estimation problems. In this paper, we study these issues in the context of the canonical Gaussian sequence problem. We propose an entirely frequentist alternative to nonparametric empirical Bayes methods by establishing a connection between simultaneous estimation and penalized nonparametric regression. We use flexible regularization strategies, such as shape constraints, to derive accurate estimators without appealing to Bayesian arguments. We prove that our estimators achieve asymptotically optimal regret and show that they are competitive with or can outperform nonparametric empirical Bayes methods in simulations and an analysis of spatially resolved gene expression data.

翻译：多个未知参数的同步估计是科学技术领域中一类广泛重要问题的核心。当前，此类问题的最优性能由非参数经验贝叶斯方法实现。然而，这些方法仍存在两大关键问题：其一，它们以贝叶斯推理解决频率学派问题，这种哲学困境导致学界对经验贝叶斯方法论持有所保留的态度；其二，其计算依赖的密度估计在某些复杂同步估计问题中变得极不可靠。本文以经典高斯序列问题为背景研究上述问题，通过建立同步估计与惩罚非参数回归之间的联系，提出一种完全频率学派的非参数经验贝叶斯方法替代方案。我们采用形状约束等灵活正则化策略，在不依赖贝叶斯论证的情况下推导精确估计量，证明所提估计量能实现渐近最优遗憾值。模拟实验与空间分辨基因表达数据分析表明，该方法与经验贝叶斯方法性能相当甚至更优。