Computing the probability of union in the n-dimensional Euclidean space for application of the multivariate quantile: p-level efficient points

As recently studied in Lee and Prékopa (2017), in the general case of Rn,n≥3, if certain conditions among the projected sets are met, the probability of union of such sets can be efficiently calculated by the modified inclusion-exclusion formula in a recursive manner. In this paper, we study with the fullest generality how to construct such suitable conditions on the partial order relation. Upon this basis we can use the modified inclusion-exclusion formulas in any case. Numerical examples are presented.