P-box

Post date: Dec 05, 2020 12:52:4 PM

In thinking about how we might generalise the definition of a p-box to include consonant p-boxes, I am just noticing that the Wikipedian Michael Hardy, my benefactor and my nemesis, changed the mathematical definition of p-box from this (below) to that (further down). I see that his change has some advantages, but I'm not sure I like it.

Is either one amenable to a new, more expansive definition?

This is an old revision of this page, at 14:49, 17 July 2019

Let 𝔻 denote the space of distribution functions on the real numbers ℝ, i.e., 𝔻 = {D | D : ℝ → [0,1], D(x) ≤ D(y) whenever x < y, for all x, y ∈ ℝ} and the limit of D at +∞ is 1 and the limit at −∞ is 0, and let 𝕀 denote the set of closed real intervals, i.e., 𝕀 = {i | i = [i₁, i₂], i₁ ≤ i₂, i₁, i₂ ∈ ℝ}. Then a p-box is a quintuple {F, F, m, v, F}, where F, F ∈ 𝔻, while m, v ∈ 𝕀, and F ⊆ 𝔻. This quintuple denotes the set of distribution functions F ∈ 𝔻 matching the following constraints:

where integrals of the form are Riemann–Stieltjes integrals.

This is an old revision of this page, at 15:00, 17 July 2019

A cumulative probability distribution function (c.d.f.) on the real numbers ℝ, is a function D : ℝ → [0,1], for which D(x) ≤ D(y) whenever x < y, and the limit of D at +∞ is 1 and the limit at −∞ is 0. A p-box is a set of cumulative distributions functions F satisfying the following constraints, for specified c.d.f.s F F, and specified numbers m₁ ≤ m₂ and v₁ ≤ v₂.