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

概率分布的形状和/或位置的某些特征的比较渗透到金融，财富分配，保险和风险的理论和应用中，仅举几个这些比较发挥基本作用的领域。然而，尽管在这个方向上的一些努力，仍然有一个很大的差距，在社会，经济和行为科学文献统计推断方法（估计和测试）与这些领域固有的分布的偏序所产生的问题。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假设所需的统计方法来弥补这一差距。非参数估计的基础分布的利益开发下各种偏序的限制和他们的渐近理论描绘使用的结果从理论的经验过程。删失数据的情况下，也被认为是，并扩展到多变量的情况下进行了讨论。作为渐近结果的副产品，提出了置信带和渐近检验，并评估了它们的操作特性，这项工作影响了各种社会问题中风险的有效评估，因此，它影响了决策过程。例如，要精确计算与不同收入不平等程度相关的死亡风险，就需要准确的建模和推理程序，以指导预防性保健措施的供资。此外，这项工作还提供了更好的工具来评估极端事件的可能性，如洪水、大风、高污染物水平等，以及这些极端事件可能发生的程度。最后，该项目从事非传统的研究生，并为他们提供了在多学科环境中的研究机会。

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.