英文标题：
《Risk aggregation with empirical margins: Latin hypercubes, empirical
copulas, and convergence of sum distributions》
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作者：
Georg Mainik
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最新提交年份：
2015
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英文摘要：
This paper studies convergence properties of multivariate distributions constructed by endowing empirical margins with a copula. This setting includes Latin Hypercube Sampling with dependence, also known as the Iman--Conover method. The primary question addressed here is the convergence of the component sum, which is relevant to risk aggregation in insurance and finance. This paper shows that a CLT for the aggregated risk distribution is not available, so that the underlying mathematical problem goes beyond classic functional CLTs for empirical copulas. This issue is relevant to Monte-Carlo based risk aggregation in all multivariate models generated by plugging empirical margins into a copula. Instead of a functional CLT, this paper establishes strong uniform consistency of the estimated sum distribution function and provides a sufficient criterion for the convergence rate $O(n^{-1/2})$ in probability. These convergence results hold for all copulas with bounded densities. Examples with unbounded densities include bivariate Clayton and Gauss copulas. The convergence results are not specific to the component sum and hold also for any other componentwise non-decreasing aggregation function. On the other hand, convergence of estimates for the joint distribution is much easier to prove, including CLTs. Beyond Iman--Conover estimates, the results of this paper apply to multivariate distributions obtained by plugging empirical margins into an exact copula or by plugging exact margins into an empirical copula.
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中文摘要：
本文研究了用copula函数赋予经验裕度构造的多元分布的收敛性。此设置包括具有相关性的拉丁超立方体采样，也称为Iman--Conover方法。这里讨论的主要问题是组成和的收敛性，这与保险和金融中的风险聚合有关。本文表明，对于聚合风险分布的CLT是不可用的，因此潜在的数学问题超出了经典的经验连接函数CLT。这一问题与所有多变量模型中基于蒙特卡罗的风险聚合有关，这些模型是通过将经验利润率插入copula生成的。本文建立了估计和分布函数的强一致相合性，并给出了概率收敛速度$O（n^{-1/2}）$的一个充分判据。这些收敛结果适用于所有密度有界的copula。具有无界密度的例子包括二元克莱顿和高斯copulas。收敛结果并不特定于分量和，对于任何其他分量非递减聚合函数也是如此。另一方面，联合分布估计的收敛性更容易证明，包括CLT。除了Iman——Conover估计之外，本文的结果还适用于通过将经验裕度插入一个精确copula或将精确裕度插入一个经验copula而获得的多元分布。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Risk Management 风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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一级分类：Statistics 统计学
二级分类：Computation 计算
分类描述：Algorithms, Simulation, Visualization
算法、模拟、可视化
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一级分类：Statistics 统计学
二级分类：Methodology 方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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