Pure-tail and Partial Orders of Probability Distributions: Theory, Inference, and Applications in Insurance, Finance, and The Social and Behavioral Sciences

通过形状和/或位置的某些特征对概率分布进行比较，渗透到金融、财富分配、保险和风险的理论和应用中，仅举几个例子，在这些领域中，这些比较起着基本的作用。然而，尽管在这个方向上做了一些努力，但在社会、经济和行为科学文献中，关于统计推断方法(估计和检验)的差距仍然很大，这些方法与这些领域固有的分布偏序有关。这项工作通过开发估计和检验金融、保险、商品捆绑、投资(投资组合选择)以及贫困和不平等衡量等领域的假设所需的统计方法，缩小了这一差距。在不同的偏序约束下，建立了感兴趣的基础分布的非参数估计，并利用经验过程理论的结果描绘了它们的渐近理论。文中还考虑了截尾数据的情形，并讨论了多元情形的推广。作为渐近结果的副产品，提出了置信度和渐近检验，并对它们的运行特性进行了评估，这项工作影响了对各种社会问题的风险的有效评估，从而影响了决策过程。例如，精确计算与不同程度的收入不平等有关的死亡风险，需要准确的建模和推论程序，以指导预防性保健措施的供资。此外，这项工作提供了更好的工具来评估极端事件发生的可能性，如洪水、大风、高水平污染物等，以及这些极端事件可能发生的水平。最后，这个项目吸引了非传统的研究生，并为他们提供了在多学科环境下的研究机会。

Comparisons of Probability distributions by certain features of their shapes and/or location permeate the theory and applications of finance, wealth distribution, insurance, and risk, to name but a few of the areas where these comparisons play a fundamental role. Nevertheless, and despite some efforts in this direction, there remains a substantial gap in the social, economic, and behavioral sciences literature regarding statistical inference methodology (estimation and testing) for problems arising in connection with the partial orders of distributions intrinsic to these areas. This work closes this gap by developing the statistical methodology needed to estimate and test hypotheses in areas such as Finance, Insurance, Commodity bundling, Investments (portfolio selection), and measurements of Poverty and Inequality, among others. Nonparametric estimators for the underlying distributions of interest are developed under various partial order restrictions and their asymptotic theory is delineated using results from the theory of empirical processes. The case of censored data is also considered, and extensions to the multivariate case are discussed. As a byproduct of the asymptotic results, confidence bands and asymptotic tests are proposed and their operating characteristics are evaluated.This work impacts the efficient evaluation of risk in various societal problems and, as a consequence, it impacts the decision -making process. For example, precise calculation of risk of mortality as it relates to various levels of income inequality requires accurate modeling and inferential procedures to guide the funding of preventive health measures. In addition, this work provides better tools to assess the likelihood of extreme events, such as floods, high winds, high levels of pollutants, etc., and the levels at which these extreme events can occur. Finally, this project engages nontraditional graduate students and provides them with research opportunities in a multidisciplinary setting.