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

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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.