AI 中文总结
研究将洛伦兹曲线扩展到多变量设置的挑战，基于条件分布和分位数函数提出向量值双变量洛伦兹曲面，建立其性质，开发非参数估计器并评估性能，通过实例说明该方法在收入不平等和精算数据中的实用性。
AI 中文摘要
洛伦兹曲线是衡量不平等的基本工具，但将其扩展到多变量设置仍具有挑战性，因为变量之间存在复杂的依赖结构，且需要捕捉不平等的方向性。本文基于条件分布和条件分位数函数引入了一种新颖的向量值双变量洛伦兹曲面（VBLS）。与现有的对称双变量洛伦兹曲面不同，所提出的VBLS有效地捕捉了两个变量之间条件依赖产生的方向性不平等。我们建立了该曲面的几个基本性质并研究了其数学性质。定义了相应的平等曲面，从而开发了相关的向量值双变量基尼测度来量化不平等。我们进一步推导了表征结果，证明了所提出的VBLS在基础分布框架内的唯一性。开发了VBLS的非参数估计器，并通过模拟研究评估了它们的有限样本性能。还通过应用于收入不平等和精算数据说明了所提出方法的实用性。
英文摘要
The Lorenz curve is a fundamental tool for measuring inequality, but its extension to multivariate settings remains challenging due to the complex dependence structure among variables and the need to capture directional aspects of inequality. In this paper, we introduce a novel vector-valued bivariate Lorenz surface (VBLS) based on conditional distributions and conditional quantile functions. Unlike existing symmetric bivariate Lorenz surfaces, the proposed VBLS effectively captures the directional inequality arising from the conditional dependence between two variables. We establish several fundamental properties of the proposed surface and investigate its mathematical properties. The corresponding egalitarian surface is defined, leading to the development of associated vector-valued bivariate Gini measures for quantifying inequality. We further derive characterization results that demonstrate the uniqueness of the proposed VBLS within the underlying distributional framework. Nonparametric estimators of the VBLS are developed and their finite-sample performance is evaluated through a simulation study. The usefulness of the proposed methodology is also illustrated with applications to income inequality and actuarial data.
