Novel transformation of unimodal (a)symmetric possibility distributions into probability distributions

Abstract

The paper introduces a new transformation of possibility distributions into continuous symmetric and asymmetric probability distributions. This transformation is called Inverse Two-Sided Normalized and could be possibly the inverse transformation of Two-Sided Normalized [1] (Foulloy et al., 2021) and the inverse construction presented in [2] (Souliotis et al., 2022). In addition, it is shown that the proposed transformation is the extension of the optimal transformation [3] (Dubois et al., 1992) and of its inverse only in specific cases. Those cases are the symmetric distributions or specific distributions such as the two-piece skewed distribution [1] (Foulloy et al., 2021). It is also shown that the asymmetry preservation of the probability distribution has 1-1 correspondence with the asymmetry preservation of the possibility distribution about the mode.