In real life situations it is common to deal with the comparison of alternatives. The alternatives to be compared are sometimes defined under some lack of information. Two lacks of information are considered: uncertainty and imprecision. Uncertainty refers to situations in which the possible results of the experiment are precisely described, but the exact result of the experiment is unknown; imprecision refers to situations in which the result of the experiment is known but it cannot be precisely described. In this work, uncertainty is modelled by means of Probability Theory, imprecision is modelled by means of IF-set Theory, and the Theory of Imprecise Probabilities is used when both lacks of information hold together.
Alternatives under uncertainty are modelled by means of random variables. Thus, a stochastic order is needed for their comparison. In this work two particular stochastic orders are considered: stochastic dominance and statistical preference. The former is one of the most usual methods used in the literature and the latter is the most adequate method for comparing qualitative variables. Some properties about such methods are investigated. In particular, although stochastic dominance is related to the expectation of some transformation of the random variables, statistical preference is related to a different location parameter: the median. In addition, some conditions, related to the copula that links the random variables, under which stochastic dominance and statistical preference are related are given. Both stochastic orders are defined for the pairwise comparison of random variables. Thus, an extension of statistical preference for the comparison of more than two random variables is defined, and its main properties are studied.
When the alternatives are defined under uncertainty and imprecision, each one is represented by a set of random variables. For comparing them, stochastic orders are extended for the comparison of sets of random variables instead of single ones. When the stochastic order is either stochastic dominance or statistical preference, the comparison of sets of random variables can be related to the comparison of elements of the imprecise probability theory, like p-boxes. Two particular instances of comparison of sets of random variables, common in decision making problems, are studied: the comparison of random variables with imprecision on the utilities or in the beliefs. The former situation is modelled by random sets, and then their sets of measurable selections are compared, and the second is modelled by a set of probabilities. When there is imprecision in the marginal distributions of the random variables, the joint distribution cannot be obtained from Sklar’s Theorem. For this reason, an imprecise version of Sklar’s Theorem is given, and its applications to bivariate stochastic orders under imprecision are showed.
Alternatives defined under imprecision, but not under uncertainty, are modeled by means of IF-sets. For their comparison a mathematical theory of comparison of IF-sets is given, focusing on a particular type of measure called IF-divergences. This measure has several applications, like for instance in pattern recognition or decision making. IF-sets are used to model bipolar information because they allow membership and non-membership degrees. Since imprecise probabilities also allow to model bipolarity, a connection between both theories is established. As an application of this connection, an extension of stochastic dominance for the comparison of more than two p-boxes is showed.