发表机构
Université Paris Cité（巴黎西岱大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究多元凸序的投影检验充分性，通过规范函数和径向反例揭示其不足，并证明对中心化高斯分布与两个高斯混合的比较，投影检验在所有有限维度下是充分且必要的。
AI 中文摘要
在精算科学和定量金融中，凸序提供了一种比较具有相同均值的风险的自然方式。在一维情形下，凸序通过多种刻画被充分理解。在更高维度中，一种自然的方法是比较所有一维投影，但尽管这一条件是必要的，它通常不足以刻画多元凸序。Pinelis 利用 Popoviciu 不等式构造了一个简单的反例。我们使用规范函数重新审视该例子，并证明即使是在二维情形下，规范函数本身也不足以刻画对称分布的多元凸序。随后，我们表明径向情形使得失败的原因变得透明，并在每个至少为二维的维度中构造了径向反例。尽管如此，投影检验对于某些重要的分布族是充分的。特别地，对于共同径向分布随机向量的线性变换，情况正是如此。该框架包括由共同径向分布生成的椭圆分布，尤其是中心化高斯分布。除了高斯分布之间的比较，Jourdain 和 Pagès 最近研究了高斯分布与高斯混合的比较，获得了必要条件与充分条件，但它们的等价性在一般情况下仍未解决。在本文中，我们解决了中心化高斯分布与两个中心化高斯分布混合的比较问题：在每个有限维度中，当且仅当所有一维投影在凸序下有序时，该高斯分布在多元凸序下被该混合分布所支配。
英文摘要
In actuarial science and quantitative finance, convex order provides a natural way to compare risks with the same mean. In dimension one, convex order is well understood through several characterisations. In higher dimensions, a natural approach is to compare all one-dimensional projections, but, although necessary, the resulting condition is in general not sufficient for multivariate convex order. A simple counterexample due to Pinelis exploits Popoviciu's inequality. We revisit this example using gauge functions, and show that gauge functions themselves do not suffice to characterise multivariate convex order for symmetric distributions, even in dimension two. We then show that the radial case makes the source of the failure transparent and construct radial counterexamples in every dimension at least two. Nevertheless, projection tests are sufficient for some important families of distributions. In particular, this is the case for linear transformations of a common radially distributed random vector. This framework includes elliptical distributions generated from a common radial distribution and, in particular, centred Gaussian distributions. Beyond comparisons between Gaussian distributions, Jourdain and Pagès recently studied the comparison of a Gaussian distribution with a Gaussian mixture, obtaining necessary conditions and sufficient conditions whose equivalence was left open in general. In this paper, we settle the comparison of a centred Gaussian distribution with a mixture of two centred Gaussian distributions: in every finite dimension, the Gaussian is dominated by the mixture in multivariate convex order if and only if all one-dimensional projections are ordered in convex order.
