A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type

Joint distributions of random sets and their relation to copulas

“…It should be noted, though, that this approach can be fruitful for particular copulas or containment functionals with certain properties as will be mentioned later. 1], i = 1, 2, be (marginal) containment functionals given by…”

“…By the Choquet theorem there exist unique probability measures i : 1] the product measure of 1 and 2 one has that…”

“…Alternatively to the approach of defining a joint containment functional by (11), one can follow an approach presented by Alvarez [1]. The latter can be applied to random sets of indexable type, i.e., random sets whose underlying probability space is the interval (0, 1] equipped with the Lebesgue measure.…”

“…The latter can be applied to random sets of indexable type, i.e., random sets whose underlying probability space is the interval (0, 1] equipped with the Lebesgue measure. It directly deals with random sets instead of their capacity (or containment) functionals and makes use of the fact that associated with a copula C one can define a measure μ C on (0, 1] 2 by [1] is to define a joint random (closed) set X from marginal random (closed) sets X 1 and X 2 . The underlying probability space is = (0, 1] 2 equipped with the measure μ C and the focal sets of X are defined by the Cartesian products X(u 1 , u 2 ) = X 1 (u 1 ) × X 2 (u 2 ).…”

“…Of course, this approach is also applicable for random sets in finite spaces E 1 , E 2 since the latter can be represented as random sets of indexable type by subdividing (0, 1] into subintervals whose lengths correspond to the probability weights of the focal sets (see [1] or [16,Example 3]). However, ψ defined in (13) depends on the order of the subdivisions of the intervals (0, 1] since μ C not only depends on the lengths of the subintervals but also on their bounds.…”