Majorisation as a theory for uncertainty

Abstract

Majorisation, also called rearrangement inequalities, yields a type of stochastic ordering in which two or more distributions can be then compared. This method provides a representation of the peakedness of probability distributions and is also independent of the location of probabilities. These properties make majorisation a good candidate as a theory for uncertainty. We demonstrate that this approach is also dimension free by obtaining univariate decreasing rearrangements from multivariate distributions, thus we can consider the ordering of two, or more, distributions with different support. We present operations including inverse mixing and maximise/minimise to combine and analyse uncertainties associated with different distribution functions. We illustrate these methods for empirical examples with applications to scenario analysis and simulations.