Essentially this work contributes to the foundations of probabilistic modeling of imprecise observed data (coarse data) and their statistical inference, with emphasis on the field of financial economics.;Our main contributions are as follows. (i) Using the theory of continuous lattices, we provide a rigorous and unified framework for modeling of random set data as bona fide random elements within probability theory, in which their distributions are characterized by Choquet capacity functionals. (ii) Turning to statistical decision theory in financial economics, we provide a general and unified approach to the popular theory of stochastic dominance, by developing approximations of utility functions, as well as investigating of statistical inference procedures. (iii) Focusing on statistical decision theory under risk in financial economics, we use the concepts of Choquet capacities and Choquet integral to investigate coherent risk measures, from which statistical inference procedures based on empirical data can be developed appropriately, such as V-statistics based upon non-parametric estimation of distribution functions.
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