标题： 幂次型损失函数的最小最大排他性类 显示英文标题

标题： Minmax Exclusivity Classes for Power-Type Loss Functions

摘要： 在统计决策理论中，损失函数的选择从根本上决定了哪些估计量可以被视为最优的。 本文引入并发展了损失函数的排他性类别的普遍概念：损失函数的子集，使得根据指定的概念，不同类别的损失函数下没有估计量可以是最优的。 我们关注的是最小最大最优性的案例，并定义了最小最大排他性类别，证明了经典的幂类型损失函数$L_p(\theta,a) = |\theta - a|^p$形成这样的类别。 在标准的正则性和光滑性假设下，我们证明了没有估计量可以同时对于属于两个不同$L_p$类别的损失函数是最小最大的。 这一结果是通过一个依赖于风险泛函可微性和损失空间锥结构的扰动论证获得的。 我们明确了排他性划分的框架，区分了平凡和可实现的结构，并分析了它们的代数性质。 这些结果开启了一个关于估计量最优性几何的更广泛研究，并探讨了通过排他性原理对损失函数空间进行分类的潜力。 显示英文摘要

摘要： In statistical decision theory, the choice of loss function fundamentally shapes which estimators qualify as optimal. This paper introduces and develops the general concept of exclusivity classes of loss functions: subsets of loss functions such that no estimator can be optimal (according to a specified notion) for losses lying in different classes. We focus on the case of minmax optimality and define minmax exclusivity classes, demonstrating that the classical family of power-type loss functions $L_p(\theta,a) = |\theta - a|^p$ forms such a class. Under standard regularity and smoothness assumptions, we prove that no estimator can be simultaneously minmax for losses belonging to two distinct $L_p$ classes. This result is obtained via a perturbation argument relying on differentiability of risk functionals and the conic structure of loss spaces. We formalize the framework of exclusivity partitions, distinguishing trivial and realizable structures, and analyze their algebraic properties. These results open a broader inquiry into the geometry of estimator optimality, and the potential classification of the loss function space via exclusivity principles.