标题： 关于正则参数流形的渐近效率理论 显示英文标题

标题： Towards an Asymptotic Efficiency Theory on Regular Parameter Manifolds

摘要： 渐近效率理论是现代数学统计学基础中的支柱之一。 它不仅作为评估统计方法的严格理论基准，还阐明了如何开发和统一新的统计程序。 例如，影响函数的微积分在过去几十年中导致了许多重要的统计突破。 为了应对分析日益复杂的数据集的紧迫挑战，特别是那些具有非欧几里得/非线性结构的数据集，近年来提出了许多新的统计模型和方法。 然而，现有的效率理论并不总是适用于这些情况，因为该理论在很大程度上是在被忽视的前提条件下发展起来的，即样本空间和参数空间都是赋范线性空间。 因此，赋范线性空间以外的效率结果非常稀少且孤立，通常是逐个案例获得的。 本文旨在发展一种更统一的渐近效率理论，允许样本空间、参数空间或两者都是满足某些正则性条件的黎曼流形。 我们构建了一个词汇表，有助于将效率理论中的基本概念从赋范线性空间转换到黎曼流形，例如（局部）正则估计量、可微泛函等。 在与赋范线性空间平行的条件下建立了效率界限。 我们还通过将其应用于统计学中的两个具体例子来展示新框架的概念优势：总体弗雷歇均值和单指标模型的回归系数向量。 显示英文摘要

摘要： Asymptotic efficiency theory is one of the pillars in the foundations of modern mathematical statistics. Not only does it serve as a rigorous theoretical benchmark for evaluating statistical methods, but it also sheds light on how to develop and unify novel statistical procedures. For example, the calculus of influence functions has led to many important statistical breakthroughs in the past decades. Responding to the pressing challenge of analyzing increasingly complex datasets, particularly those with non-Euclidean/nonlinear structures, many novel statistical models and methods have been proposed in recent years. However, the existing efficiency theory is not always readily applicable to these cases, as the theory was developed, for the most part, under the often neglected premise that both the sample space and the parameter space are normed linear spaces. As a consequence, efficiency results outside normed linear spaces are quite rare and isolated, obtained on a case-by-case basis. This paper aims to develop a more unified asymptotic efficiency theory, allowing the sample space, the parameter space, or both to be Riemannian manifolds satisfying certain regularity conditions. We build a vocabulary that helps translate essential concepts in efficiency theory from normed linear spaces to Riemannian manifolds, such as (locally) regular estimators, differentiable functionals, etc. Efficiency bounds are established under conditions parallel to those for normed linear spaces. We also demonstrate the conceptual advantage of the new framework by applying it to two concrete examples in statistics: the population Frechet mean and the regression coefficient vector of Single-Index Models.